AAPL wrote:Manhattan Prep

The positive difference of the fourth powers of two consecutive positive integers must be divisible by

A. One less than twice the larger integer.
B. One more than twice the larger integer.
C. One less than four times the larger integer.
D. One more than four times the larger integer.
E. One more than eight times the larger integer.

OA A

Using, 1 and 2, we have 1 and 16, so the difference is 15 (which is divisible by 3).

Using 2 and 3, we have 16 and 81, so the difference is 65 (which is divisible by 5).

Using 3 and 4, we have 81 and 256, so the difference is 175 (which is divisible by 7).

We see that in each scenario the results must be divisible by one less than twice the larger integer.

Alternate solution:

Since all the answer choices involves the larger integer, we can let n = the larger integer. Thus n - 1 = the smaller integer and the positive difference between their fourth powers is:

n^4 - (n - 1)^4

[n^2 - (n - 1)^2][n^2 + (n - 1)^2]

[n^2 - (n^2 - 2n + 1)][n^2 + (n^2 - 2n + 1)]

(2n - 1)(2n^2 - 2n + 1)

Since it has a factor of 2n - 1, it must be divisible by 2n - 1, i.e., one less than twice the larger integer.

Answer: A

AAPL wrote: ↑

Thu Oct 17, 2019 5:41 am

Manhattan Prep

The positive difference of the fourth powers of two consecutive positive integers must be divisible by

A. One less than twice the larger integer.
B. One more than twice the larger integer.
C. One less than four times the larger integer.
D. One more than four times the larger integer.
E. One more than eight times the larger integer.

OA A

I would probably opt for testing values for this question

Start with consecutive integers 1 and 2
1⁴ = 1 and 2⁴ = 16
Positive difference = 16 - 1 = 15

Now check the answer choices...

(A) one less than twice the larger integer
2(2) - 1 = 3
15 is divisible by 3
KEEP A

(B) one more than twice the larger integer
2(2) + 1 = 5
15 is divisible by 5
KEEP B

(C) one less than four times the larger integer
4(2) - 1 = 7
15 is NOT divisible by 7
ELIMINATE C

(D) one more than four times the larger integer
4(2) + 1 = 9
15 is NOT divisible by 9
ELIMINATE D

(E) one more than eight times the larger integer
8(2) + 1 = 17
15 is NOT divisible by 17
ELIMINATE E

We're left with A and B
Test another pair of consecutive integers

How about 2 and 3
2⁴ = 16 and 3⁴ = 81
Positive difference = 81 - 16 = 65

(A) one less than twice the larger integer
2(3) - 1 = 5
65 is divisible by 5
KEEP A

(B) one more than twice the larger integer
2(3) + 1 = 7
65 is NOT divisible by 7
ELIMINATE B

Answer: A

Cheers,
Brent