enigma123 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 2Options for n:
1, 3, 5, 7, 9, 11, 13...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(1+1)(1-1)}{24} = \frac{0}{24} =\) 0 R0
If n=3, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(3+1)(3-1)}{24} = \frac{8}{24} =\) 0 R8
Since R can be different values, INSUFFICIENT.
Statement 2: n is not divisible by 3Options for n:
1, 2, 4, 5, 7, 8...
If n=1, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(1+1)(1-1)}{24} = \frac{0}{24} =\) 0 R0
If n=2, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(2+1)(2-1)}{24} = \frac{3}{24} =\) 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:
\(\frac{(1+1)(1-1)}{24} = \frac{0}{24} =\) 0 R0
If n=5, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(5+1)(5-1)}{24} = \frac{24}{24} =\) 1 R0
If n=7, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(7+1)(7-1)}{24} = \frac{48}{24} =\) 2 R0
If n=11, then dividing (n+1)(n-1) by 24 yields the following:
\(\frac{(11+1)(11-1)}{24} = \frac{120}{24} =\) 5 R0
In every case, R=0.
SUFFICIENT.
.
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 nSince 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.
.
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