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
OE C
Well I can get the answer by taking numbers but can any1 explain it practically.
Remainder
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Statement 1: n is not divisible by 2akash singhal 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
Case 1: n=1
Here, (n-1)(n+1)/24 = (0*2)/24 = 0/24 = 0 R0.
Case 2: n=3
Here, (n-1)(n+1)/34 = (2*4)/24 = 3/24 = 0 R3.
Since different remainders are possible, INSUFFICIENT.
Statement 2: n is not divisible by 3
Case 1 also satisfies Statement 2.
In Case 1, R=0.
Case 3: n=4
Here, (n-1)(n+1)/34 = (3*5)/24 = 15/24 = 0 R15.
Since different remainders are possible, INSUFFICIENT.
Statements combined:
n is equal to an ODD integer NOT DIVISIBLE BY 3:
1, 5, 7, 11, 13...
As shown above, n=1 results in R=0.
Case 4: n=5
Here, (n-1)(n+1)/34 = (4*6)/24 = 24/24 = 1 R0.
Case 5: n=7
Here, (n-1)(n+1)/34 = (6*8)/24 =48/24 = 2 R0.
In every case, R=0.
SUFFICIENT.
The correct answer is C.
Conceptual explanation for the combined statements:
n-1, n, n+1 are 3 consecutive integers.
Of every 3 consecutive integers, exactly 1 will be a multiple of 3.
Since n is NOT divisible by 3, either n-1 or n+1 must be a multiple of 3.
Since n = odd, n-1 and n+1 are two consecutive EVEN integers.
Of every two consecutive even integers, exactly one will be a multiple of 4.
Thus, (n-1)(n+1) = (multiple of 4)(even non-multiple of 4) = multiple of 8.
Since (n-1)(n+1) is divisible by both 3 and 8, (n-1)(n+1) is a multiple of 24.
Thus, dividing (n-1)(n+1) by 24 will yield a remainder of 0.
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