Math Revolution Approach (PS)

[GMAT math practice question]

If r is an integer, and 700 < 10!/(10-r)!<1,000, then r=

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


=>

Note that
10!/(10-r)! = 10 * (10-1) * … * (10-r+1).

Test different values of r:
r=1 : 10
r=2 : 10 * 9 = 90
r=3 : 10 * 9 * 8 = 720
r=4 : 10 * 9 * 8 * 7 = 5040

Therefore, r = 3.

The answer is C.

Answer: C

[GMAT math practice question]

i, j, and k are non-negative integers such that i+j+k=3. If p, q, and r are three fixed, but different, prime numbers, how many different values of p^iq^jr^k are possible?

A. 8
B. 9
C. 10
D. 11
E. 12

=>

The number of possible values of p^iq^jr^k is equal to the number of solutions of the equation i + j + k = 3.
The solution set of the equation i + j + k = 3 includes all permutations of (3,0,0), (2,1,0), and (1,1,1).
The number of permutations of (3,0,0) is 3!2! = 3.
The number of permutations of (2,1,0) is 3! = 6.
The number of permutations of (1,1,1) is 1.

Therefore, the number of solutions of the equation i+j+k=3 is 3 + 6 + 1 = 10.

Therefore, the answer is C.

Answer: C

[GMAT math practice question]

If (x+2)!/x! = 72, then x=

A. 6
B. 7
C. 8
D. 9
E. 10

=>
Now,
72= (x+2)!/x! = [x!*(x+1)*(x+2)]/x! =(x+1)(x+2).

So,
x2 + 3x + 2 = 72
x2 +3x – 70 = 0
(x-7)(x+10) = 0
x = 7 or x = -10.

Since factorials are only defined for x ≥0, x = 7.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

A circle with center (6,8) and radius 2 lies in the standard xy-plane. What is the shortest distance between the origin and the points of the circle?

A. 4
B. 6
C. 8
D. 10
E. 12



The distance between the origin and the center of the circle is 10=√(6^2+4^2).
Since the radius of the circle is 2, the point of the circle closest to the origin lies along the line joining the center to the origin, and so has distance 10 – 2 = 8 from the origin.

Therefore, the answer is C.

Answer : C

[GMAT math practice question]



In the number line above, the ticks are evenly spaced. Which point represents 2^{10}?

A. A
B. B
C. C
D. D
E. E

=>

The distance between each pair of consecutive ticks is 2^9 – 2^8 = 2^8(2-1) = 2^8.
Find the values of each point:
A = 2^9 + 2^8.
B = A + 2^8 = 2^9 + 2^8 + 2^8 = 2^9 + 2*2^8 = 2^9 + 2^9 = 2*2^9 = 2^{10}.

Therefore, the answer is B.

Answer : B

[GMAT math practice question]

For positive integers x and y, x@y is defined by x@y=(x+y)/xy. If a, b, and c are positive integers, what is the value of 1/a@1/(1/b@1/c)?

A. a+b+c
B. 1/abc
C. 1/(a+b+c)
D. 1/(ab+bc+ca)
E. 3/abc

=>

We can simplify the definition of the operation in the following way:
x@y=(x+y)/xy = x/(xy) + y/(xy) = 1/x+1/y.
So,
1/a@1/(1/b@1/c) = a + (1/b @1/ c) = a + ( b + c ).

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

Nov. 18th fell on a Thursday in 1999. On which day did Nov. 18th fall in 2005?

A. Tuesday
B. Wednesday
C. Thursday
D. Friday
E. Saturday

=>

If a year is divisible by 400, it is a leap year.
If a year is divisible by 100, but not divisible by 400, it is not a leap year.
If a year is divisible by 4, but not divisible by 100, it is a leap year.

The day on which a particular date falls will be shifted by two days from one year to the next if the next year is a leap year, and by one day from one year to the next if the next year is not a leap year. For example, Nov. 18th fell on a Thursday in 1999, and a Saturday in 2000 since 2000 was a leap year. It fell on a Saturday in 2000, and a Sunday in 2001 since 2001 was not a leap year.

As there were two leap years (2000 and 2004) between 1999 and 2005, Nov. 18th shifted by 8 days over the 6-year period. If a date is shifted by 7 days, it will fall on the same day of the week. So, the net effect was to shift Nov. 18th by one day. Therefore, Nov 18th fell on a Friday in 2005, and the answer is D.

Answer: D

[GMAT math practice question]

If m>n>0, x=m^2+n^2, and y=2mn, what is the value of √x^2-√y^2 in terms of m and n?

A. m^2+n^2
B. 2m^2+n^2
C.m^2+2n^2
D. n^2-m^2
E. m^2-n^2

=>
Plugging in the expressions for x and y, and expanding gives:
x^2 – y^2 = (m^2+n^2)^2 – (2mn)^2 = m^4 -2m^2n^2 + n^4 = (m^2-n^2)^2.
So, since m^2 > n^2,
√x^2-√y^2=√(m^2-n^2)^2=|m^2-n^2|=m^2-n^2.

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

When 2 fair dice are tossed, what is the probability that the difference between the 2 numbers that land face up will be 3?

A. 1/6
B. 1/3
C. 1/2
D. 2/3
E. 5/6

=>

6 pairs of numbers with this property can appear on the dice: (1,4), (4,1), (2,5), (5,2), (3,6) and (6,3). The total number of outcomes from rolling two dice is 36.
Thus, the probability that the two numbers will have a difference of 3 is 6/36 = 1/6.

Therefore, the answer is A.
Answer : A

[GMAT math practice question]

How many real numbers x satisfy ||x-3|-4|=4?

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

=>

| |x-3| - 4 | = 4
⇔ |x-3| - 4= ± 4
⇔ |x-3| = 4 ± 4
⇔ |x-3| = 8 or |x-3| = 0
⇔ x-3 = ±8 or x-3 = 0
⇔ x = 3 ± 8 or x = 3
⇔ x = 11, x = -5 or x = 3

The equation has 3 solutions.

Therefore, the answer is C.

Answer : C

[GMAT math practice question]

Which of the following is the closest to 11^59^5 – 2(10^5)?

A. 10^2
B. 10^7
C. 10^8
D. 10^9
E. 10^{10}

=>

11^59^5 – 2(10^5)
≒ 10^510^5 – 2(10^5)
= 10^{10} – 2(10^5)
≒ 10^{10}

Therefore, the answer is E.

Answer: E

[GMAT math practice question]

What is the remainder when 7^{30} is divided by 100?

A. 19
B. 29
C. 39
D. 49
E. 59


=>

The tens digit of 7^k cycles through 0-->4-->4--->0--->0-->…..
The units digit of 7^k cycles through 7-->9-->3--->1--->7-->…..

Since 30=4(7)+2, 7^{30}=7^{4(7)+2} ends in 49.

Therefore, the answer is D.

Answer: D

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[GMAT math practice question]

If n is the sum of the first 40 positive integers, what is the greatest prime factor of n?

A. 11
B. 29
C. 31
D. 41
E. 53

=>

n = 1 + 2 + … + 40 = 40∙41/2 = 20∙41 = 2^2∙5∙41.
Thus the greatest prime factor of n is 41.
Therefore, the answer is D.

Answer: D

[GMAT math practice question]

When n is divided by 5, the remainder is 3, and when n is divided by 6, the remainder is 3. What is the remainder when the smallest possible integer value of n is divided by 7?

A. 0
B. 1
C. 2
D. 3
E. 4

=>
Plugging in numbers is the recommended approach for remainder questions.
The integers that have a remainder of 3 when they are divided by 5 are 3, 8, 13, 18, 23, 28, 33, 38, …
The integers that have a remainder of 3 when they are divided by 6 are 3, 9, 15, 21, 27, 33, 39, 38, …
The smallest integer n that occurs in both lists is 38.
Since 38 = 7 ∙5 + 3, this value of n has a remainder of 3 when it is divided by 7.
Therefore, the answer is D.

Answer : D

[GMAT math practice question]



If the distances between consecutive ticks in the above number line are the same, which of the following points represents 2^11?

A. A
B. B
C. C
D. D
E. E


=>
Let d be the distance between consecutive ticks. Then
2d = 2^10 – 2^9 = 2*2^9 – 2^9 = 2^9
d=2^8.
Since
2^11 – 2^10 = 2^32^8 – 2^22^8 = 8(2^8)– 4(2^8)= 4(2^8)= 4d,
2^11 is the fourth point from 2^10, which is D.

Therefore, the answer is D.
Answer : D

[GMAT math practice question]



Points A, B, C, and D lie on the number line as shown in the figure above. If AC=BD and AB=BC/5, what is the value of C?

A. 7/14
B. 8/14
C. 9/14
D. 10/14
E. 11/14

=>

Suppose d is the distance between A and B. Then
CD = d and BC = 5d.
Thus, AD = d + 5d + d = 7d. Since A = ½ and D = 2/3,
7d = 2/3 – 1/2 = 4/6 – 3/6 = 1/6,
and
d = 1/42.
So, D – C = 1/42, and
C = 2/3 – 1/42 = 28/42 – 1/42 = 27/42 = 9/14.

Therefore, the answer is C.

Answer: C