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Terms in this set (113)
Concerning 'sum of squares' questions, which numbers are always important to consider?
0 and 1 (Question #2 on CAT2)
0^2 = 0
1^2 = 1
What is a critical numbers property as it relates to negative numbers raised to an exponent?
-1 raised to any even exponent is 1 and that -1 raised to any odd exponent is -1
Which of the following cannot yield an odd integer when divided by 5?
a) an odd integer squared
b) an integer less than 5
c) the product of an odd integer and an even integer
(d) the sum of three consecutive integers
(e) an even integer minus an odd integer
eliminate on sight: A, B, and D
e) 10-5 = 5/5 = 1
c) 2 x 5 = 10/5 = 2 (correct answer)
LESSON: Any integer multiplied by an even integer will always produce another even integer. An even integer (like 10) divided by an odd integer (like 5), will yield an even integer (like 2).
If a question asks for the circulation for the years 1982-1990, how many years is this?
9 (NOT 8)
1990-1982 = 8 + 1 = 9
what is an effective strategy for picking numbers when a question asks whether a variable is an integer?
Pick both integer and non-integer values for variables
ex: if c and are positive, is d an integer?
statement 1) c = d^3
If c = 0.125, then d = 0.5. If c = 8, d = 2. Since d can be either an integer or a non-integer, statement (1) is INSUFFICIENT.
(ctnd. from previous flashcard)
(2) d = (radical c)
simplified: c = d^2
If c = 0.25, d = 0.5. If c = 4, d = 2. Since d can be either an integer or a non-integer, Statement (2) is INSUFFICIENT.
Combined: If c = d^3 and c = d^2, then d^3 = d^2 .
The only numbers for which this can be true are 0 and 1, and since d must be positive, the only permissible value for d is 1.
Therefore, our statements taken together are Sufficient.
Answer Choice (C) is the correct answer.
LESSON: the number ZERO is not positive -or- negative
If x < 0, yz does not equal 0, and xy^2z^3 < 0, then which of the following must be true?
I. -1 - z < 0
II. (x/z) > 0
III. (y/z) < 0
since xy^2z^3 is negative...y^2, a nonzero number taken to an even exponent, will always be positive, then exactly one of x or z^3 is negative. The question stem tells us that x < 0, so z^3, and therefore z, must be positive, since nonzero numbers taken to an odd exponent retain their sign. Our information about the variables can be summed up as follows: x is negative, y can be positive or negative, and z is positive.
I. -1 - z < 0 is the only one that can be true.
Which quant question # from CAT 2 should you have tattooed in your brain?
Question 32
The sum of two numbers is even in which two cases?
if both of those numbers are even -or- both of those numbers are odd
i.e. 3 + 3 = 6 or 4 + 2 = 6
"x^4 is divisible by 16" is another way of saying what?
16 is a factor of x^4
Strategize:
A class of n students, where n > 2, has an average (arithmetic mean) grade of G, where G > 0. A student, whose grade is only half the average, drops the class. What is the new class average?
(CAT 3, Q 30)
Abstract questions can be intimidating, but the principles behind solving them remain the same. Averages are always (total value of all entities)/(number of entities). The fact that there are variables in the Answer Choices should immediately make us think about possibly Picking Numbers.
a divisor is the same thing as what?
a factor
an integer is divisible by 3 if...
the SUM of the integer's digits is divisible by 3
an integer is divisible by 4 if...
the integer is divisible by 2 TWICE or if the last TWO digits are divisible by 4
i.e. 456 is divisible by 4 but 678 is not
an integer is divisible by 5 if...
the integer ends in ZERO or 5
an integer is divisible by 6 if...
the integer is divisible by BOTH 2 and 3
i.e. 48 is divisible by 6
an integer is divisible by 8 if...
the integer is divisible by 2 THREE TIMES -or- if the LAST THREE digits are divisible by 8.
i.e. 456 is divisible by 8 but 556 is not
an integer is divisible by 9 if...
the SUM of the integer's digits are divisible by 9.....
i.e. 4185 is divisible by 9 but 3459 is not
if you add or subtract two multiples of a number, you will get...
another multiple of the same number
i.e. 35 + 21 = 56 (also a multiple of 7)
i.e. 35 - 21 = 14 (also a multiple of 7)
what are the first 10 prime numbers?
2, 3, 5, 7, 11, 13, 17, 19, 23 and 29
what is the Factor Foundation Rule?
if A is a factor of B, and B is a factor of C, then A is a factor C
i.e. if 72 is divisible by 12, then 72 is also divisible by all of the factors of 12 (1, 2, 3, 4, 6, and 12)
is 1 or zero a prime number?
NO
given that integer n is divisible by 3, 7, and 11, what other numbers must be divisors of n?
3 x 7 = 21
3 x 11 = 33
7 x 11 = 77
21 x 11 = 231
how do you calculate the GCF and LCM of 30 and 24 using prime factorization?
30 - 2 x 3 x 5
24 - 2 x 2 x 2 x 3
GCF = product of overlapping primes = 2 x 3 = 6
LCM = product of ALL primes excluding overlap = 2 x 3 x 5 x 2 x 2 = 120
If you add or subtract 2 odds or 2 evens, what is the result?
the result is EVEN
i.e. 7 + 11 = 18
i.e. 8 + 4 = 12
if you add or subtract an ODD with an EVEN, what is the result?
the result is ODD
i.e. 7 + 8 = 15
i.e. 18 - 7 = 11
if you multiply integers, if ANY of the integers is EVEN, what is the result?
the result is EVEN
i.e. 3 x 8 x 9 = 216
if you multiply ODD integers, what is the result?
the result is ODD
i.e. 3 x 5 x 7 = 105
If you multiply several EVEN integers together (alone or with odd integers), the result will be higher and higher powers of two.
For example, if you multiply TWO even integers like 2 x 5 x 6, the result will be divisible by what power of 2?
2 x 5 x 6 = 60 which is divisible by the second power of 2 or 2^2 = 4.......60/4 = 15
Same thing with 2 x 5 x 6 x 10 = 600 which is divisible by the 3rd power of 2 or 2^3 = 8.......600/8 = 75
what are the 3 special properties of divisibility?
even divided by an even can be even or odd:
i.e. 12/6 = 2 or 12/4 = 3
even divided by an odd is even:
i.e. 12/3 = 4
odd divided by an odd:
i.e. 27/9 = 3 or 9/3 = 3
concerning sum of primes, what are the only two possible situations if you sum two primes?
a) the sum of two prime numbers will always be even and the product will always be odd (UNLESS one of the primes is the number 2)
-or-
b) if the sum of the primes is odd or the product is even, 2 must be one of the prime numbers
what are the 3 properties of EVENLY SPACED SETS?
1) smallest or largest number in the set
2) the increment (always 1 for consecutive integers)
3) the number of items in the set
4) the mean -AND- median of the set are equal to the AVERAGE of the FIRST and LAST terms
5) the SUM of the elements in the set equals the MEAN times the number of ITEMS in the set
what is the rule and formula for counting consecutive INTEGERS inclusively?
"Add one before you are done"
i.e. How many integers are there from 14 to 765?
765 - 14 = 751 + 1 = 752 (last - first + 1)
what is the formula for counting consecutive MULTIPLES inclusively?
(last - first) / increment + 1
how many multiples of 7 are there between 100 and 150?
Find one easy multiple (like 140) and and add the increment (in this case 7) to get to one of the last multiples (140 + 7 = 147)....then subtract the highest multiple to get to the other endpoint (147 - 42 = 105)....
finally, (147 - 105) / 7 = 6 (+1) = 7
what is the sum of all the integers from 20 to 100 inclusive?
1) average the first and last term to find the precise 'middle of the set': 100 + 20 = 120, 120 / 2 = 60
2) count the number of terms inclusive: 100-20 = 80, 80 + 1 = 81
3) multiply the 'middle' number by the number of terms to find the sum: 60 x 81 = 4,860
what is the Factor Foundation Rule?
*The PRODUCT of n consecutive integers is divisible by (n!)
*The SUM of n consecutive integers is divisible by n ONLY if n is odd, but it is NOT divisible by n if n is even.
Strategize:
Both 5^2 and 3^3 are factors of n × 2^5 × 6^2 × 7^3 where n is a positive integer. What is the smallest possible positive value of n?
the question must contain two 5's and three 3's.
prime factorize anything the unprime units:
n x 2^5 x 2 x 3 x 2 x 3 x 7^3
that accounts for two 3's so the smallest value of n is 3 x 5^2 = 75
What's a good strategy with picking numbers with averages questions containing variables in the answer choices?
Pick all the terms in the group to have the same value will greatly reduce the arithmetic load, which means a smaller chance of careless error.
1/4 $ 1/5 = 1/20
0 $ 5 = 0
$ could be multiplication or subtraction in the first equation:
1/4 x 1/5 = 1/20 -or- 5/20 - 4/20 = 1/20
$ could be multiplication or division in the second equation:
0 / 5 = 0
0 x 5 = 0
what's a property of SUMS of CONSECUTIVE integers?
If there are an ODD number of consecutive integers, the sum of all the integers will ALWAYS be a MULTIPLE of the number of items.
i.e. 4 + 5 + 6 + 7 + 8 = 30 (a multiple of 5)
i.e. 13 + 14 + 15 + 16 + 17 = 75 (also a multiple of 5)
If there are an EVEN number of consecutive integers, the sum of all the integers will NEVER be a MULTIPLE of the number of items.
2 + 3 + 4 + 5 = 14 (NOT a multiple of 4)
if x is an EVEN integer, is x(x+1)(x+2) divisible by 4?
if x is an EVEN integer, is x(x+1)(x+2) divisible by 4?
YES because this deals with MULTIPLICATION and the question stem says x is EVEN so x + 2 must also be EVEN which means that the product of those two or these three integers is divisible by 4 (divisible by 2 twice).
What is the sum of all positive integers up to 100 inclusive?
What is the sum of all positive integers up to 100 inclusive?
Add one before you're done: 100 - 1 = 99, 99 + 1 = 100 terms
Find the mean: 100 + 1 = 101, 101/2 = 50.5
Multiple # of terms x the mean: 100 x 50.5 = 5,050
In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?
In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?
1) Pick numbers: 8 + 7 + 6 + 5 = 26, 1 + 2 + 3 + 4 = 10, 26-10 = 16
2) 4 (integers) x 4 (the difference - between 1 and 5, 2 and 6, 3, and 7, 4 and 8) = 16
How many terms are there in the set of consecutive integers from -18 to 33, inclusive?
33 - (-18) = 51, 51 + 1 = 52
If r, s, and t are consecutive positive multiples of 3, is rst divisible by 27, 54, or both?
3 x 3 x 3 x 2 (every other multiple of 3 has a prime factor of 2) = 54
54 is divisible by 54 and 27, so the answer is both.
is the sum of the integers from 54 to 153, inclusive, divisible by 100?
153 - 54 = 99, 99 + 1 = 100. 100 is an EVEN number, so the answer is NO.
Lesson: If there are an EVEN number of consecutive integers, the sum of all the integers will NEVER be a MULTIPLE of the number of items.
what is the most important consideration when you see an EVEN exponent?
the base can be positive OR negative:
x^2 = 16, x could be 4 or -4
an odd exponent always keeps the sign of the base:
x^3 = 8, x = 2
x^3 = -8, x = -2
if x^6 = x^7 = x^15, what can x be?
if x^6 = x^7 = x^15, what can x be?
0 or 1 since there's odd and even exponents
if x^6 = x^8 = x^10, what can x be?
if x^6 = x^8 = x^10, what can x be?
0 or 1 OR -1 because all the exponents are EVEN
if x^5 = x^7 = x^11, what can x be?
if x^5 = x^7 = x^11, what can x be?
0 or 1 OR -1 because all the exponents are ODD
what's the fundamental property of positive proper fractions (fractions between 0 and 1) raised to an exponent?
as the exponent increases, the fraction decreases:
(1/2)^2 = (1/2) x (1/2) = 1/4
the same can be said of decimals raised to an exponent:
(0.6)^2 = 0.36
what are the important rules regarding simplifying a compound base raised to an exponent?
i.e. (2 x 5)^2 or (2 + 5)^2
Multiplication - either multiply the compound base or distribute the exponent to each number in the base:
(2 x 5)^2 = 10^2 = 100 -or- (2 x 5)^2 = 2^2 x 5^2 = 100
Addition: you CANNOT distribute the exponent to each number in the base; you must add:
(2 + 5)^2 = 7^2 = 49
INCORRECT: 2^2 + 5^2 = 29 (WRONG)
what is the rule for when multiplying two terms with the same base?
i.e. (3^4) x (3^2) = ?
Add:
(3^4) x (3^2) = ?
(3^4) x (3^2) = 3^(4+2) = 3^6 = 729
what is the rule for when dividing two terms with the same base?
i.e. (3^6) / (3^2) = ?
Subtract:
(3^6) / (3^2) = ?
3^(6-2) = 3^4
what is the rule for simplifying 'nested exponents'?
i.e. (3^2)^4?
Multiply:
(3^2)^4?
3^(2 x 4) = 3^8
what is the rule for negative exponents?
i.e. 5^-1
i.e. 1/(4^-2)
i.e. (-2)^-3
Put the term containing the exponent in the opposite fractional place (numerator or denominator) and make the exponent positive:
5^-1 = 1/(5^1) = 1/5
1/(4^-2) = 4^2 = 16
(-2)^-3 = 1/(-2^3) = 1/8
Complete the following sentence: Any number that does not have an exponent implicitly has...
...an exponent of 1.
5 = 5^1
8 = 8^1
Lesson: when you see an exponents question containing a base or bases without an exponent notated, write in an exponent of 1 for these base(s).
what is the rule for non-zero bases raised to an exponent of zero?
i.e. 5^0?
It always equals 1:
5^0 = 1
3^0 = 1
8^0 = 1
complete the following sentence: fractional exponents are the link between...
...roots and exponents.
in a fractional exponent, the NUMERATOR tells us...
i.e. 25^(3/2)
which POWER to raise the base to:
25^(3/2) = radical 25^3
in a fractional exponent, the DENOMINATOR tells us...
i.e. 4^(3/2)
which ROOT to take:
4^(3/2) = radical (2) of 4^3
what are the 3 main rules for simplifying EXPONENTIAL expressions?
1. You can only simplify exponential expressions linked by MULTIPLICATION or DIVISION. (you cannot simplify those dealing with addition and subtraction)
2. You can simplify exponential expressions linked by multiplication -or- division IF they have either a BASE or an EXPONENT in common.
3. Exponential expressions involving ADDITION -or- SUBTRACTION can be FACTORED but no simplified:
i.e. (7^3) + (7^7) = 7^3 (1 + 7^4)
what is the only time a root can have a negative result?
if:
1. it is an odd root (cube root, 5th root, 7th root, etc.)
2. the base of the root is negative
i.e. cube root of -27 = -3
what is the rule for SIMPLIFYING ROOTS?
you can only simplify roots by separating or combining them in MULTIPLICATION -or- DIVISION; you cannot do so with addition or subtraction.
what's a good estimation of the square root of 70?
it should be between 8 and 8.5 since the square root of 64 is 8 and the square root of 81 is 9.
70 is closer in proximity to 64 than it is to 81 so the safe bet is that the square root of 70 is less than 8.5 and more than 8.
If positive integer x is multiple of 8 and the positive integer y is a multiple of 12, then xy must be multiples of what numbers?
8 x 12 = 96
96 is divisible by 1, 2, 3, 4, 6, 8, 12, etc.
If p is the product of the integers 1 to 32, inclusive, what is the greatest integer k for which 2^k is a factor of p?
If p is the product of the integers 1 to 32, inclusive, what is the greatest integer k for which 2^k is a factor of p?
2 - 2 16 - 2x2x2x2
4 - 2x2 18 - 2x9
5 - 2x5 20 - 2x2x5
6 - 2x3 22 - 2x11
8 - 2x2x2 24 - 2x2x2x3
10 - 2x5 26 - 2x13
12 - 2x2x3 28 - 2x2x7
14 - 2x7 30 - 2x15
32 - 2x2x2x2x2
Count up two's: 31
what is the lowest positive integer that is divisible by each of the integers 1 through 8, inclusive?
2 = 2
3 = 3
4 = 2x2
5 = 5
6 = 2x3
7 = 7
8 = 2x2x2
2x2x2 (8 contains the factors of 2 for 2, 4, and 6)
x3 (6 contains the factor of 3 for 3)
x5x7 = 840
which of the following CANNOT yield an integer when divided by 10?
a) sum of two odd integers
b) an integer less than 10
c) the product of two primes
d) the sum of three consecutive integers
e) an odd integer
e) 25/10 = 2.5
if positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
1) x is a multiple of 9
2) x is a multiple of 25
prime factors of 105: 3, 5, 7
prime factors of x - 2, 3
prime factors of y - 2, 7
looking for a prime factor of 5 then:
1) ins (adds a 2nd prime factor of 3; irrelevant)
2) suf (adds a prime factor of 5 which determines that xy is a multiple of 105)
if N is an integer, then N is divisible by how many positive integers?
1) n is the product of two different prime numbers
2) n and 2^3 are each divisible by the same number of positive integers
1) n = pq, so n is divisible by 1, p, q, and pq (SUF)
2) SUF
D is the answer (either statement sufficient)
If Y is an integer, is y^3 divisible by 9?
1) y is divisible by 4
2) y is divisible by 6
1) y being divisible by 4 indicates that y has two prime factors of the number 2 (INS)
2) y being divisible by 6 indicates that y has prime factors of 2 and 3, so y^3 would be divisible by 9 (and 27)
is MPT even if m < p < t and all are integers?
t + m = 2p
maybe, maybe not:
t + m = 2p
23 + 7 = 2 (15)
4 + 6 = 2 (5)
if r, s, and t are non-zero integers, is (r^5)(s^3)(t^4) negative?
1) rt is negative
2) s is negative
if r, s, and t are non-zero integers, is (r^5)(s^3)(t^4) negative?
1) r or t is negative (INS)
2) s is negative (INS)
T) t^4 is negative, s^3 is negative; unclear about r^5 though (INS)
is N an integer?
1) n^2 is an integer
2) sq root N is an integer
1) INS: (1.414)^2 is an integer (2) even though N is not an integer
2) SUF
What is the value of POSITIVE integer N?
1) N^4 < 25
2) N does not equal N^2
What is the value of POSITIVE integer N?
1) N^4 < 25
N could be 2, 1 (INS)
2) N does not equal N^2
N is not 1 (INS)
T) N is 2 and not 1; SUF
list all the prime numbers up to 100 (write them out and then check your work on the back of this flashcard)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 27, 71, 73, 79, 83, 89, 97
If you add a multiple of N to a non-multiple of N, the result is....
If you add a multiple of N to a non-multiple of N, the result is....
...a non-multiple of N. (same holds true for subtraction)
7 + 3 = 10 (not a multiple of 7 or 3)
7 - 3 = 4 (not a multiple of 7 or 3)
If you add two non-multiples of N, the result could either be...
If you add two non-multiples of N, the result could either be...
...a multiple of N or a non-multiple of N.
19 + 13 = 32 (non multiple of 3)
19 + 14 = 33 (multiple of 3)
is N divisible by 7?
1) N = x - y, where x and y are integers
2) x is divisible by 7, and y is not divisible by 7
is N divisible by 7?
1) N = x - y, where x and y are integers
doesn't tell us if x or y is divisible by 7 or anything about N (INS)
2) x is divisible by 7, and y is not divisible by 7
doesn't tell us anything about N (INS)
T) x is a multiple of 7 and y is not a multiple of 7 where N = x - y. The difference between x and y can NEVER be divisible 7. (SUF)
what is the general properties for calculating GCF?
it is calculated by multiplying the SHARED prime factors.
i.e. GCF of 108 and 27
prime factors of 108 = 2 x 2 x 3 x 3 x 3
prime factors of 27 = 3 x 3 x 3
GCF: 3 x 3 x 3 = 27
what is the general properties for calculating LCM?
it is calculated by multiplying the highest exponential factors:
i.e. LCM of 12 and 9:
prime factors of 12: 2 x 2 x 3
prime factors of 9: 3 x 3
LCM: 3^2 x 2^2 = 36
is the integer z divisible by 6?
1) the GCF of z and 12 is 3
2) the GCF of z and 15 is 15
is the integer z divisible by 6?
IOW: does z have prime factors of 2 and 3?
1) the GCF of z and 12 is 3
since the GCF is 3 (and the GCF contains no prime factor of 2 -AND- because 12 contains two prime factors of 2), z is NOT divisible by 6 (SUFFICIENT)
2) the GCF of z and 15 is 15
since the GCF of z and 15 contains prime factors of 3 and 5 but doesn't mention anything about a 2, this is INSUFFICIENT as 15 does not contain any 2s, but z could.
what do all perfect squares (i.e. 16, 25, 36, etc) have in common as it relates to total factors?
all perfect squares have an ODD number of total factors
i.e. 36 (1, 2, 3, 4, 6, 9, 12, 18, 36) has a total of 9 factors
what do non-perfect squares (8, 12, 18, etc.) all have in common as a relates to total factors?
all non-perfect squares have an EVEN number of total factors
i.e. 12 (1, 2, 3, 4, 6, 12) has a total of 6 factors
what special property do perfect squares have as it relates to their prime factorization?
what special property do perfect squares have as it relates to their prime factorization?
the prime factorization of a perfect square contains only EVEN POWERS of primes
i.e. 90^2 = (2 x 3^2 x 5) (2 x 3^2 x 5) = (2^2)(3^4)(5^2)
how many different factors does 2000 have?
prime factorize:
2^4 x 5^3 = 2000
count the exponent individually (4 and 3) and add 1 to each before you're done (4 + 1)(3 + 1) and then multiply: 5 x 4 = 20 different factors
there are 20 different factors for the number 2000
a, b, and c are positive integers > 1. if a < b < c and abc = 286, what is c-b?
prime factorize:
286/2 = 143, 143/13 = 11
286's prime factors (and only factors besides 1 and 286 are 2, 11, and 13. Since a, b, and c must be > 1, a = 2, b = 11, and c = 13. so b - c = 2.
if z is an integer and z! is divisible by 340, what is the smallest possible value for z?
prime factorize:
340 = 10 (2 x 5) and 34 (17 x 2)
in order for z! to be divisible by 340, it must be divisible by all the prime factors of 340. in order for z! to be divisible by 2, z must be a least 2; in order for z! to be divisible by 5, z must be at least 5, etc. therefore, in order for z! to be divisible by 17, z must be at least 17. 17 is the smallest possible value for z if z! is divisible by 340.
which of the following numbers has exactly 15 factors?
105
108
120
144
168
perfect squares are the only numbers that have an ODD number of factors, so you can eliminate everything but 144 (judicious laziness)
to check though, prime factorize 144 = 2^4 x 3^2
take the exponents of the prime factors and add 1 to each before you're done and multiply: (4 + 1) (2 + 1) = 5 x 3 = 15
is p divisible by 168?
1) p is divisible by 14
2) p is divisible by 12
is p divisible by 168? p would require prime factors of 2^3, 3, and 7 to be sufficient
1) p is divisible by 14; prime factors of 2 and 7 (INS)
2) p is divisible by 12; prime factors of 2^2 and 3 (INS)
T) p is divisible by 2^2, 3, and 7 (INS). answer is E.
DO NOT count a redundant 2 to indicate that p has three prime factors of 2.
what is the GCF of x and y?
1) x and y are both divisible by 4
2) x - y = 4
what is the GCF of x and y?
1) x and y are both divisible by 4 (INS); x could = 16 and y could = 8, x could = 8 and y = 4
2) x - y = 4 (INS), x could = 17, y could = 13; x could = 18, y could = 14; both pairs of #s have different GCFs
T) x and y are multiples of 4 and they are 4 apart from eachother so they are consecutive multiples of 4.
LESSON: property of GCFs: consecutive multiples of N have a GCF of N
if x^2 is divisible by 216, what is the smallest possible value for x?
prime factorize 216: 2^3 and 3^3
organize prime factors and x^2 into two columns of x:
x x
2 2
3 3
2
3
since x^2 is a perfect square, the prime factors must be paired. because there are odd exponents of the prime factors, organize them as indicated above. smallest value will be the product of 2 x 3 x 2 x 3 = 36
what is the 'length' of an integer?
total number of prime factors
what is the maximum length of any number less than 600?
take the smallest prime number (2) to maximize length:
2^6 = 64
64 x 2 = 128
128 x 2 = 256
256 x 2 = 512 (2^9)
so max length of an integer < 600 is 9
if x and y are positive integers and x/y has a remainder of 7, what is the smallest possible value of xy?
RULE: remainder will always be smaller than divisor
so 7 must be smaller than y, so y must be at least 8. if y is 8, then the smallest possible value for x must be 7 (0/8 = 0 remainder 7). other possible values for x could be 15 (15/8 = 1 remainder 7), 23, 31, 39, etc.
smallest value of xy (7 x 8) = 56
when 15 is divided by y, the remainder is y - 3. If y must be an integer, what are all the possible values for y?
RULE: remainders must be non-negative
Algebra is faster than picking numbers bro:
15 = (x + y) + y - 3
18 = xy + y
18 = y (x + 1)
therefore, y must be a factor of 18, so y could be 1, 2, 3, 6, 9, or 18. remainder must be non-negative so y must be 3, 6, 9, or 18 (y - 3 doesn't work with 1 or 2).
what properties of remainders must you consider when looking at odd and even integers?
-Odd integers are those that leave a remainder of 1 after division of 2
-Even integers are those that leave a remainder of 0 after division of 2
Represented algebraically:
Even: 2n
Odd: 2n + 1 or 2n - 1
what is the best strategy for solving absolute value questions that contain two different variables?
solve conceptually by picking numbers and -NOT- algebraically
what is the best strategy for solving absolute value questions that contain one variable and one or more constants?
solve algebraically rather than picking numbers/using a conceptual approach
if x, y, and z are prime numbers and x < y < z, what is the value of x?
1) xy is even
2) xz is even
x must be 2
answer D; either statement is sufficient
if C and D are integers, is C - 3D even?
1) C and D are odd
2) C - 2D is odd
if C and D are integers, is C - 3D even? NTK: need info about C and D to be sufficient.
1) C and D are odd; ODD - ODD = even always (SUF)
2) C - 2D is odd; ODD - EVEN = ODD but this tells us nothing about D (INS)
answer is A; statement 1 is sufficient
A machinist's salary at a factory increases by $2,000 at the end of each fully year the machinist works. If the machinist's salary for the 5th year is $39,000, what is the machinist's average annual salary for his first 21 years in the factory?
FORMULA: to find the mean of evenly spaced sets:
avg. salary = (first yr + last yr) / 2
$39,000 - 4 yrs ($2,000) = $31,000 (1st year)
$39,000 + 16 ($2,000) = $71,000 (21st year)
$71k + $31k = $102k / 2 = $51,000 avg salary for first 21 yrs
Is the average of N consecutive integers equal to 1?
1) N is even
2) If S is the sum of the N consecutive integers, then 0 < S < N.
Is the average of N consecutive integers equal to 1?
1) N is even; the mean of an EVEN number of consecutive integers will NEVER be an integer (SUF).
2) If S is the sum of the N consecutive integers, then 0 < S < N; simplify my multiplying everything by N:
0/N < (S/N) < N/N
0 < (S/N) < 1 (SUF). answer is D (either statement is suffice)
The product 7 x 6 x 5 x 4 x 3 x 2 x 1 is divisible by all of the following except:
a) 120
b) 240
c) 360
d) 840
e) 1260
step 1) prime factorize 7 x 6 x 5 x 4 x 3 x 2 x 1 = 2^3, 3^2, 5, 7...so correct answer would have some extraneous # of prime factors.
step 2) prime factorize answers:
120 = 10 x 12 = 2 x 5 x 2 x 2 x 3 (eliminate)
240 = 10 x 24 = 2 x 5 x 2 x 2 x 2 x 3 (one extra 2 here; so this is the answer)
the only time you can combine or subtract roots is...
if they have the same sign under the radical:
(sq root 80) - (sq root 45)
sq root 5 x 16 = 4 sq root 5
sq root 9 x 5 = 3 sq root 5
different = sq root 5
when attempting to simplify a fraction with a root in the denominator involving addition or subtraction....you should do what?
multiply both top and bottom by the CONJUGATE of the denominator:
for A + sq root B, the conjugate = A - sq root B and vice versa
see p. 166 of Manhattan Number Properties book for example on this.
when positive integer x is divided by y, the remainder is 9. if x/y = 96.12, what is the value of y?
9.00/0.12 = tens digit of 7 (9 x 7 = 84) and units digit of 5 (90-84 = 6)
75 x .12 = 9
If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be?
A) 2
B) 5
C) 6
D) 7
E) 14
factorize 3150: comes out to 2 x 3^2 x 5^2 x 7
to be a perfect square 3150y must have an even number of each of its prime factors. in this case, you would need an additional 2 and 7 or 2 x 7 = 14 (answer E)
What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out?
Find GCF of 91 and 21:
91 = 7 x 13
21 = 7 x 3
GCF is 7; so 7 bouquets of 13 red and 3 white tulips each is the largest number of identical bouquets that can be made leaving no flowers left over.
if X is an integer and Y = 3X + 2, which of the following cannot be a divisor of Y?
a) 4
b) 5
c) 6
d) 7
e) 8
Although 3X is always divisible by 3, 3x + 2 cannot ever be divisible by 3 since 2 is not divisible by 3. Thus, 3x + 2 cannot be any multiple of 3, including 6 (answer C)
A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value of (10^4 - 10^2)(0.0012) with the second term having digits 12 repeat?
Distribute:
(10^4)(0.0012) - (10^2)(0.0012) =10k(0.0012) - 100 (0.0012)
move decimal point a number of spaces equal to the number of zeros in the first terms:
10k(0.0012) = 12.12 -and- 100(0.0012) = 0.1212
12.12 (repeating) - 0.12 (repeating = 12
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