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If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest
prime factor of k? - ✔✔Consecutive Integers:
Thus, since there are no multiples of 30 between 295 and 299 and between 601 and 615, finding the
sum of all multiples of 30 between 295 and 615, inclusive, is equivalent to finding the sum of all
multiples of 30 between 300 and 600, inclusive. Therefore, we can rephrase the question: "What is the
greatest prime factor of the sum of all multiples of 30 between 300 and 600, inclusive?"
The sum of a set = (the mean of the set) × (the number of terms in the set)
Since 300 is the 10th multiple of 30, and 600 is the 20th multiple of 30, we need to count all multiples
of 30 between the 10th and the 20th multiples of 30, inclusive.
There are 11 terms in the set: 20th - 10th + 1 = 10 + 1 = 11
The mean of the set = (the first term + the last term) divided by 2: (300 + 600) / 2 = 450
k = the sum of this set = 450 × 11
Note, that since we need to find the greatest prime factor of k, we do not need to compute the actual
value of k, but can simply break the product of 450 and 11 into its prime factors:
k = 450 × 11 = 2 × 3 × 3 × 5 × 5 × 11
Therefore, the largest prime factor of k is 11.
The total cost of producing item X is equal to the sum of item X's overhead cost and production cost. If
the production cost of producing X decreased by 5% in January, by what percent did the total cost of
producing item X change in that same month?
(1) The overhead cost of producing item X increased by 13% in January.
,(2) Before the changes in January, the overhead cost of producing item X was 5 times the production
cost of producing item X. - ✔✔Percents:
Answer: C
If the square root of p^2 is an integer greater than 1, which of the following must be true?
I. p^2 has an odd number of positive factors
II. p^2 can be expressed as the product of an even number of positive prime factors
III. p has an even number of positive factors
a) I
b) II
c) III
d) I and II
e) II and III - ✔✔Divisibility of Primes:
Statement I: 36's factors can be listed by considering pairs of factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6). We
can see that they are 9 in number. In fact, for any perfect square, the number of factors will always be
odd. This stems from the fact that factors can always be listed in pairs, as we have done above. For
perfect squares, however, one of the pairs of factors will have an identical pair, such as the (6,6) for 36.
The existence of this "identical pair" will always make the number of factors odd for any perfect square.
Any number that is not a perfect square will automatically have an even number of factors. Statement I
must be true.
Statement II: 36 can be expressed as 2 x 2 x 3 x 3, the product of 4 prime numbers.
A perfect square will always be able to be expressed as the product of an even number of prime factors
because a perfect square is formed by taking some integer, in this case 6, and squaring it. 6 is comprised of
one two and one three. What happens when we square this number? (2 x 3)2 = 22 x 32. Notice that each
prime element of 6 will show up twice in 62. In this way, the prime factors of a perfect square will always
appear in pairs, so there must be an even number of them. Statement II must be true.
,Statement III: p, the square root of the perfect square p2 will have an odd number of factors if p itself is
a perfect square as well and an even number of factors if p is not a perfect square. Statement III is not
necessarily true.
The correct answer is D, both statements I and II must be true.
Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A
and B work together at their respective rates to complete the job, which of the following represents the
fraction of the job that B will not have to complete?
a) (x-y)/(x+y)
b) x/(y-x)
c) (x+y)/xy
d) y/(x-y)
e) y/(x+y) - ✔✔Rates & Work:
We can solve this problem as a VIC (Variable In Answer Choice) and plug in values for the two variables,
x and y. Let's say x = 2 and y = 3.
Machine A can complete one job in 2 hours. Thus, the rate of Machine A is 1/2.
Machine B can complete one job in 3 hours. Thus, the rate of Machine B is 1/3.
The combined rate for Machine A and Machine B working together is: 1/2 + 1/3 = 5/6.
Using the equation (Rate)(Time) = Work, we can plug 5/6 in for the combined rate, plug 1 in for the total
work (since they work together to complete 1 job), and calculate the total time as 6/5 hours.
The question asks us what fraction of the job machine B will NOT have to complete because of A's help.
In other words we need to know what portion of the job machine A alone completes in that 6/5 hours.
, If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the
value of x?
(1) z is prime
(2) x is prime - ✔✔Exponents & Roots:
The best way to answer this question is to use the exponential rules to simplify the question stem, then
analyze each statement based on the simplified equation.
(3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8)(7^10)(3^28)(x^y) Divide both sides by common terms 5^8, 7^10, 3^27
(5^2)(z) = 3x^y
(1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of
3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have
another factor besides the 3. Therefore z = 3.
Since z = 3, the left side of the equation is 75, so x^y = 25. The only integers greater than 1 that satisfy
this equation are x = 5 and y = 2, so statement (1) is sufficient. Put differently, the expression x^y must
provide the two fives that we have on the left side of the equation. The only way to get two fives if x
and y are integers greater than 1 is if x = 5 and y = 2.
(2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5
to balance out the 5^2 on the left side. Since statement (2) says that x is prime, x cannot have any other
factors, so x = 5. Therefore statement (2) is sufficient.
If $ defines a certain operation, is p $ q less than 20?
(1) x $ y = 2x2 - y for all values of x and y
Exams + OG Quant| Complete
Verified Answers
If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest
prime factor of k? - ✔✔Consecutive Integers:
Thus, since there are no multiples of 30 between 295 and 299 and between 601 and 615, finding the
sum of all multiples of 30 between 295 and 615, inclusive, is equivalent to finding the sum of all
multiples of 30 between 300 and 600, inclusive. Therefore, we can rephrase the question: "What is the
greatest prime factor of the sum of all multiples of 30 between 300 and 600, inclusive?"
The sum of a set = (the mean of the set) × (the number of terms in the set)
Since 300 is the 10th multiple of 30, and 600 is the 20th multiple of 30, we need to count all multiples
of 30 between the 10th and the 20th multiples of 30, inclusive.
There are 11 terms in the set: 20th - 10th + 1 = 10 + 1 = 11
The mean of the set = (the first term + the last term) divided by 2: (300 + 600) / 2 = 450
k = the sum of this set = 450 × 11
Note, that since we need to find the greatest prime factor of k, we do not need to compute the actual
value of k, but can simply break the product of 450 and 11 into its prime factors:
k = 450 × 11 = 2 × 3 × 3 × 5 × 5 × 11
Therefore, the largest prime factor of k is 11.
The total cost of producing item X is equal to the sum of item X's overhead cost and production cost. If
the production cost of producing X decreased by 5% in January, by what percent did the total cost of
producing item X change in that same month?
(1) The overhead cost of producing item X increased by 13% in January.
,(2) Before the changes in January, the overhead cost of producing item X was 5 times the production
cost of producing item X. - ✔✔Percents:
Answer: C
If the square root of p^2 is an integer greater than 1, which of the following must be true?
I. p^2 has an odd number of positive factors
II. p^2 can be expressed as the product of an even number of positive prime factors
III. p has an even number of positive factors
a) I
b) II
c) III
d) I and II
e) II and III - ✔✔Divisibility of Primes:
Statement I: 36's factors can be listed by considering pairs of factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6). We
can see that they are 9 in number. In fact, for any perfect square, the number of factors will always be
odd. This stems from the fact that factors can always be listed in pairs, as we have done above. For
perfect squares, however, one of the pairs of factors will have an identical pair, such as the (6,6) for 36.
The existence of this "identical pair" will always make the number of factors odd for any perfect square.
Any number that is not a perfect square will automatically have an even number of factors. Statement I
must be true.
Statement II: 36 can be expressed as 2 x 2 x 3 x 3, the product of 4 prime numbers.
A perfect square will always be able to be expressed as the product of an even number of prime factors
because a perfect square is formed by taking some integer, in this case 6, and squaring it. 6 is comprised of
one two and one three. What happens when we square this number? (2 x 3)2 = 22 x 32. Notice that each
prime element of 6 will show up twice in 62. In this way, the prime factors of a perfect square will always
appear in pairs, so there must be an even number of them. Statement II must be true.
,Statement III: p, the square root of the perfect square p2 will have an odd number of factors if p itself is
a perfect square as well and an even number of factors if p is not a perfect square. Statement III is not
necessarily true.
The correct answer is D, both statements I and II must be true.
Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A
and B work together at their respective rates to complete the job, which of the following represents the
fraction of the job that B will not have to complete?
a) (x-y)/(x+y)
b) x/(y-x)
c) (x+y)/xy
d) y/(x-y)
e) y/(x+y) - ✔✔Rates & Work:
We can solve this problem as a VIC (Variable In Answer Choice) and plug in values for the two variables,
x and y. Let's say x = 2 and y = 3.
Machine A can complete one job in 2 hours. Thus, the rate of Machine A is 1/2.
Machine B can complete one job in 3 hours. Thus, the rate of Machine B is 1/3.
The combined rate for Machine A and Machine B working together is: 1/2 + 1/3 = 5/6.
Using the equation (Rate)(Time) = Work, we can plug 5/6 in for the combined rate, plug 1 in for the total
work (since they work together to complete 1 job), and calculate the total time as 6/5 hours.
The question asks us what fraction of the job machine B will NOT have to complete because of A's help.
In other words we need to know what portion of the job machine A alone completes in that 6/5 hours.
, If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the
value of x?
(1) z is prime
(2) x is prime - ✔✔Exponents & Roots:
The best way to answer this question is to use the exponential rules to simplify the question stem, then
analyze each statement based on the simplified equation.
(3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8)(7^10)(3^28)(x^y) Divide both sides by common terms 5^8, 7^10, 3^27
(5^2)(z) = 3x^y
(1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of
3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have
another factor besides the 3. Therefore z = 3.
Since z = 3, the left side of the equation is 75, so x^y = 25. The only integers greater than 1 that satisfy
this equation are x = 5 and y = 2, so statement (1) is sufficient. Put differently, the expression x^y must
provide the two fives that we have on the left side of the equation. The only way to get two fives if x
and y are integers greater than 1 is if x = 5 and y = 2.
(2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5
to balance out the 5^2 on the left side. Since statement (2) says that x is prime, x cannot have any other
factors, so x = 5. Therefore statement (2) is sufficient.
If $ defines a certain operation, is p $ q less than 20?
(1) x $ y = 2x2 - y for all values of x and y
