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# Math Revolution Approach (PS)

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14 Jun 2018, 01:47
[GMAT math practice question]

When n is divided by 2, 3, 5, 7 and 9, the remainder is 1. What is the smallest value of the positive integer n?

A. 100
B. 211
C. 421
D. 631
E. 841

=>

Since 2, 3, 5 and 7 are prime numbers, n = 2 ∙3 ∙5 ∙7 ∙k + 1 = 210 ∙k + 1 for some positive integer k.
If k = 1, then n = 210 ∙1 + 1 = 211 has remainder 4 when it is divided by 9 since 211 = 9 ∙23 + 4.
If k = 2, then n = 210 ∙2 + 1 = 421 has remainder 7 when it is divided by 9 since 421 = 9 ∙46 + 7.
If k = 3, then n = 210 ∙3 + 1 = 631 has remainder 1 when it is divided by 9 since 631 = 9 ∙70 + 1.

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15 Jun 2018, 01:05
[GMAT math practice question]

N is the 36th prime number. N has remainder 1 when it is divided by 3 and it has remainder 1 when it is divided by 5. What is the remainder when N is divided by 2?

A. -1
B. 0
C. 1
D. 2
E. 3

=>

All prime numbers but 2 are odd integers.
Thus, the remainder when N is divided by 2 is 1.

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17 Jun 2018, 17:28
[GMAT math practice question]

x and y are integers such that 4x^2 – y^2 +4x + 4y – 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number

=>

4x^2 – y^2 +4x + 4y – 3 = 0
=> 4x^2 + 4x + 1 – y^2 + 4y – 4 = 0
=> 4x^2 + 4x + 1 = y^2 - 4y + 4
=> (2x+1)^2 = (y-2)^2
Since 2x + 1 is an odd integer, (y-2)^2 is an odd integer and y-2 is an odd integer.
Thus, y is also an odd integer.

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17 Jun 2018, 17:29
[GMAT math practice question]

If n is the product of the squares of 4 different prime numbers, how many factors does n have?

A. 8
B. 16
C. 27
D. 64
E. 81

=>

n = p^2q^2r^2s^2 where p, q, r and s are 4 different prime numbers.
Then the number of factors of n is (2+1)(2+1)(2+1)(2+1) = 34 = 81.

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20 Jun 2018, 00:47
[GMAT math practice question]

If the average of five positive integers is 16, and the largest of the integers is 40, then the median of the five integers could be which of the following?

I 10
II 15
III 20

A. I only
B. II only
C. III only
D. I & II only
E. I, II & III only

=>

Since the average of the five positive integers is 16, their sum is 5*16 = 80. So, the four smallest numbers must sum to 80 – 40 = 40.

As the integers are positive, the smallest possible median occurs if the integers are 1,1,1,37 and 40. The largest possible median occurs if the third and fourth integers are as large as possible. For this to occur, the two smallest integers must be as small as possible, that is, 1 and 1. In this case, the two remaining integers add to 40 – (1 + 1) = 38. The largest possible median occurs if the two remaining integers are equal to 38/2 = 19. Therefore, the median lies between 1 and 19, inclusive. So, 20 is not the median.

If the numbers are 1,1,10,28 and 40, the median is 10, and if the numbers are 1,1,15, 23 and 40, the median is 15.

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21 Jun 2018, 00:44
[GMAT math practice question]

What is the sum of the remainders when the first 30 positive integers are divided by 5?

A. 50
B. 55
C. 60
D. 65
E. 70

=>

1, 6, 11, 16, 21 and 26 have remainder 1 when they are divided by 5.
2, 7, 12, 17, 22 and 27 have remainder 2 when they are divided by 5.
3, 8, 13, 18, 23 and 28 have remainder 3 when they are divided by 5.
4, 9, 14, 19, 24 and 29 have remainder 4 when they are divided by 5.
5, 10, 15, 20, 25 and 30 have remainder 0 when they are divided by 5.

The sum of the remainders is
1*6 + 2*6 + 3*6 + 4*6 + 0*6 = ( 1 + 2 + 3 + 4 + 0 ) * 6 = 60.

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21 Jun 2018, 23:46
[GMAT math practice question]

If the product of 5 different positive integers is less than 200, what is the greatest possible value for the largest of these 5 integers?

A. 3
B. 6
C. 7
D. 8
E. 9

=>
The greatest possible value of the largest of the integers occurs when the other integers are as small as possible.
In other words, the other four integers should be 1,2,3, and 4.

We try different values for the fifth integer:
1*2*3*4*5 = 120
1*2*3*4*6 = 144
1*2*3*4*7 = 168
1*2*3*4*8 = 192
1*2*3*4*9 = 216

The largest product that is less than 200 occurs when the fifth integer is 8.
Thus, the greatest possible value of the largest integer is 8.

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24 Jun 2018, 18:18
[GMAT math practice question]

When a, b and c are consecutive positive even integers such that a>b>c, which of the following must be an odd integer?

A. (a-c)/2
B. (c-a)/2
C. (a+c)/2
D. (a+c)/4
E. (a-c)/4

=>

Write a = 2n + 2, b = 2n and c = 2n – 2. We check each of the alternatives.

A. ( a – c ) / 2 = ( 2n + 2 – ( 2n – 2 ) ) / 2 = 4 / 2 = 2.
B. ( c – a ) / 2 = ( 2n – 2 – ( 2n + 2 ) ) / 2 = -4 / 2 = -2.
C. ( a + c ) / 2 = ( 2n + 2 + 2n – 2 ) / 2 = 4n / 2 = 2n.
D. ( a + c ) / 4 = ( 2n + 2 + 2n – 2 ) / 4 = 4n / 4 = n.
E. ( a – c ) / 4 = ( 2n + 2 – ( 2n – 2 ) ) / 4 = 4 / 4 = 1

Only option E is guaranteed to be odd. Therefore, the answer is E.
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24 Jun 2018, 18:19
[GMAT math practice question]

Three jars contain the same numbers of balls. 2 balls from the first jar are moved to the second jar and 4 balls from the first jar are moved to the third jar. If the ratio between the numbers of balls in the second and third jars after moving the balls is 13 to 14, how many balls were originally in each jar?

A. 12
B. 15
C. 21
D. 24
E. 30

=>

Suppose each jar originally contained n balls.
After moving balls from the first jar to the second and third jars, the second and the third jars contain n + 2 and n + 4 balls, respectively.
So,
n+2 : n+4 = 13 : 14
=> 13(n+4) = 14(n+2)
=> 13n + 52 = 14n + 28
=> n = 24
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27 Jun 2018, 01:01
[GMAT math practice question]

Alice and Bob drive the same car. Alice drives the car for 1/4 of one trip, and Bob drives the car for the remainder of the same trip. The difference between the distance driven by Bob and the distance driven by Alice is 40 km. What is the total distance travelled on the trip?

A. 60 miles
B. 80 miles
C. 100 miles
D. 120 miles
E. 125 miles

=>

Let d be the total distance traveled on the trip.
Alice drives (1/4)d and Bob drives (3/4)d.
The difference between the distances they drive is (3/4)d – (1/4)d = (1/2)d = 40.
Thus, d = 80.

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28 Jun 2018, 02:36
[GMAT math practice question]

3^n+3^{n+1}+3^{n+2}/3^{n-2}+3^{n-1}+3^n=?

A. 3^{-2}
B. 3^{-n}
C. 3^n
D. 3^2
E. 1

=>

(3^n + 3^{n+1} + 3^{n+2}) / (3^{n-2} + 3^{n-1} + 3^n)
= (3^n(1 + 3^1 + 3^2)) / (3^{n-2}(1 + 3^1 + 3^2 ))
= (3^{2+(n-2)}*13) / (3^{n-2}*13)
= 3^2

We can also find the answer by plugging in n = 2.
This gives (3^2 + 3^3 + 3^4)/(3^0 + 3^1 + 3^2) = (9 + 27 + 81)/(1 + 3 + 9) = 107/13 = 9 = 3^2.

In addition, we can plug-in number 2.
Then we have (32 + 33 + 34)/(30 + 31 + 32) = (9+27+81)/(1+3+9) = 107/13 = 9.

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29 Jun 2018, 00:18
[GMAT math practice question]

If An+1=120 + (1/4)An for each positive integer n, and A1=150, which of these ranges of values includes A12?

A. 110-120
B. 120-130
C.130-140
D. 140-150
E. 150-160

=>

A1 = 150, A2 = 120 + (1/4)150 = 120 + 32.5 = 152.5. As An is increasing,
A12 must lie between 150 and 160.

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01 Jul 2018, 17:13
[GMAT math practice question]

693 and n have the same prime factors and n is a multiple of 693 that is greater than 693. What is the smallest possible value of n/693?

A. 2
B. 3
C. 5
D. 7
E. 11

=>

Since 693 = 3^2*7*11, the smallest integer multiple of 693 that is greater than 693 with prime factors 3, 7 and 11 only is n = 693*3.
Thus, n / 693 = ( 693 * 3 ) / 693 = 3.

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01 Jul 2018, 17:15
[GMAT math practice question]

If Alice traveled on a 10 km trip at a constant speed of 25 km/h and a 48 km trip at a constant speed of 30 km/h, what was her average speed, in kilometers per hour, for the two trips?

A. 26 km/h
B. 28 km/h
C. 29 km/h
D. 30 km/h
E. 32 km/h

=>

The total time taken for the two trips is 10 / 25 + 48 / 30 = 0.4 + 1.6 = 2 hours.
The total distance traveled on the two trips is 10 + 48 = 58 km.
Thus, the average speed for the two trips is the total distance over the total time, or 58 / 2 = 29 km / h.

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04 Jul 2018, 00:54
[GMAT math practice question]

Water enters a cylindrical barrel at a constant speed of 500 cm^3/min, and the height of the barrel increases at a constant speed of 10 cm per minute. What is the approximate radius of the barrel, in centimeters?

A. 1
B. 2
C. 3
D. 4
E. 5

=>

Let r be the radius of the barrel.
The area of the water surface 3.14*r^2.
The volume of water poured in 1 minute is 10*3.14*r^2.
Then, 10*3.14*r^2 = 500 or 31.4*r^2 = 500.
r^2 = 500 / 31.4 ≒ 16.

Thus, the radius is approximately 4 cm.

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04 Jul 2018, 23:51
[GMAT math practice question]

Which of the following equations has a graph that can pass through only one coordinate (x,y) in which both x and y are integers?

A. y=2x
B. y=√2*x+1
C. y= 1/x^2 - 1
D. y=1/x^3
E. y=(-1/x)+1

=>

A. x = 1, y = 2 and x = 2, y = 4 lie on this graph.
B. Since √2 is irrational, the only point on this graph with integer coordinates is x = 0, y = 1.
C. x = 1, y = 0 and x = -1, y = 0 lie on this graph.
D. x = 1, y = 1 and x = -1, y = -1 lie on this graph.
E. x = 1, y = 0 and x = -1, y = 2 lie on this graph.

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06 Jul 2018, 00:45
[GMAT math practice question]

In the xy-plane, line L has equation y=mx+b, where 1<m<5. If line L passes through (1,1), which of the following points also lies on line L?

A. (2, -1)
B. (2,0)
C. (2,1)
D. (2,2)
E. (2,3)

=>

When we plug in the values x = 1 and y = 1, we obtain 1 = m + b or b = 1 – m.
All answer choices have x-coordinates of 2. Plugging 2 in for the variable x yields
y = 2m + b = 2m + 1 – m = m + 1.
Since 1 < m < 5, we must have 2 < m + 1 < 6 and 2 < y < 6.
Thus, (2,3) is the only point that can lie on the line.

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08 Jul 2018, 03:19
[GMAT math practice question]

The relationship between the weight W in pounds and the height H in inches of an object is given by the equation, W = 110 +5*(H-60). If the height is increased by 10 inches, by how much does the weight increase?

A. 30 pounds
B. 40 pounds
C. 50 pounds
D. 60 pounds
E. 70 pounds

=>

From W = 110 +5*(H-60), we have W = 5H – 190, which is the equation of a line of slope 5. Since the slope is the ratio of the increase in the weight to that of the height, the weight is increased by 50 when the height is increased by 10.

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08 Jul 2018, 03:21
[GMAT math practice question]

John invested 1/4 of A dollars in an investment fund, and placed the remaining 3/4 of A dollars in a savings account. If he made a 4p% loss on the fund, and a p% profit on the savings account, what was his total profit (or loss), in terms of A and p? (A negative value represents a loss.)

A. –Ap/400
B. 0
C. Ap/400
D. Ap/200
E. Ap/100

=>
Total profit = profit on investment fund + profit on savings account
= (1/4)A*(-4p/100) + (3/4)A*(p/100)
=-4Ap/400 + 3Ap/400 = -Ap/400

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11 Jul 2018, 00:11
[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III

=>

Statement I
7(3n+1) = 21n + 7 = 3(7n+2) + 1
Thus, [3n+1] = 1

Statement II
7(3n) = 21n = 3*7n + 0
Thus, [3n] = 0

Statement III
Since [3n] = 0, we have [3n] + 2 = 2.

Thus, only [3n+1] equals 1.

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Re: Math Revolution Approach (PS)   [#permalink] 11 Jul 2018, 00:11

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# Math Revolution Approach (PS)

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