binaras wrote:
If "n" is a positive integer and "r" is the remainder when (n-1)(n+1) is divided by 24.
What is the value of "r"?
1. n is not divisible by 2
2. n is not divisible by 3
Statement 1: n is not divisible by 2
Options for n:
1, 3, 5, 7, 9, 11, 13...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=3, then dividing (n+1)(n-1) by 24 yields the following:
(3+1)(3-1) / 24 = 8/24 = 0 R8.
Since R can be different values, INSUFFICIENT.
Statement 2: n is not divisible by 3
Options for n:
1, 2, 4, 5, 7, 8...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=2, then dividing (n+1)(n-1) by 24 yields the following:
(2+1)(2-1) / 24 = 3/24 = 0 R3.
Since R can be different values, INSUFFICIENT.
Statements combined:
Options for n:
1, 5, 7, 11...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
(1+1)(1-1) / 24 = 0/24 = 0 R0.
If n=5, then dividing (n+1)(n-1) by 24 yields the following:
(5+1)(5-1) / 24 = 24/24 = 1 R0.
If n=7, then dividing (n+1)(n-1) by 24 yields the following:
(7+1)(7-1) / 24 = 48/24 = 2 R0.
If n=11, then dividing (n+1)(n-1) by 24 yields the following:
(11+1)(11-1) / 24 = 120/24 = 5 R0.
In every case, R=0.
SUFFICIENT.
The correct answer is
C.
Alternate approach:
Statement 1: 2 is not a factor of n.
Thus, n = odd.
Thus, (n-1)(n+1) = the product of two consecutive even integers.
Of every two consecutive even integers, exactly one is a multiple of 4.
Thus, the product of 2 consecutive even integers = the product of an even integer and a multiple of 4 = a multiple of 8.
Since a multiple of 8 can be a multiple of 24 (in which case r=0) or not be a multiple of 24 (in which case r≠0), INSUFFICIENT.
Statement 2: 3 is not a factor of n
Since one of every 3 consecutive integers is a multiple of 3, and n is not a multiple of 3, either (n-1) or (n+1) must be a multiple of 3.
Thus, (n-1)(n+1) = a multiple of 3.
If (n-1)(n+1) is also a multiple of 8, then (n-1)(n+1) = a multiple of 24, in which case r=0.
If (n-1)(n+1) is not a multiple of 8, then (n-1)(n+1) ≠a multiple of 24, in which case r≠0.
INSUFFICIENT.
Statements 1 and 2 combined:
Since (n-1)(n+1) = a multiple of 8, and either n-1 or n+1 must be a multiple of 3, (n-1)(n+1) = a multiple of 24.
When a multiple of 24 is divided by 24, r=0.
SUFFICIENT.
The correct answer is
C.
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