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第二题是选C
9. 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.
Sol:
ST 1- if n is not divisible by 2, then n is odd, so both(n - 1) and (n + 1) are even. Moreover, since every other even number is a multiple of 4, one of those two factors is a multiple of 4. So the product (n -1)(n + 1) contains one multiple of 2 and one multiple of 4, so it contains at least 2 x 2 x 2 = three 2's in its prime factorization.
But this is not sufficient, because it can be (n-1)*(n+1)can be 2*4 where remainder is 8. it can be 4*6 where the remainder is 0.
ST 2- if n is not divisible by 3, then exactly one of (n- 1) and (n + 1) is divisible by 3, because every third integer is divisible by 3. Therefore, the product (n - 1)(n + 1) contains a 3 in its prime factorization.
Just like st 1 this is not sufficient
ST1+ST2 - the overall prime factorization of (n - 1)(n +1) contains three 2's and a 3.
Therefore, it is a multiple of 24.
Sufficient
Answer C
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