Re: If n is a positive integer, is n^2 - 1 divisible by 24?
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05 Sep 2020, 12:22
Statement (1) tells us that n is a prime number.
If n = 2, the lowest prime number, then n2 – 1 = 4 – 1 = 3, which is not divisible by 24.
If n = 3, the next prime number, then n2 – 1 = 9 – 1 = 8 which is not divisible by 24.
However, if n = 5, then n2 – 1 = 25 – 1 = 24, which is divisible by 24.
Thus, statement (1) alone is not sufficient.
Now let us examine statement (2). By design, this number is large enough so that it would not be easy to
check numbers directly. Thus, we need to go straight to number properties.
For an expression to be divisible by 24, it must be divisible by 2, 2, 2, and 3 (since this is the prime
factorization of 24). In other words, the expression must be divisible by 2 at least three times and by 3 at least
once.
The expression n2 – 1 = (n – 1)(n + 1).
If we think about 3 consecutive integers, with n as the middle number, the expression n2 – 1 is the product of
the smallest number (n – 1) and the largest number (n + 1) in the consecutive set.
Given 3 consecutive positive integers, the product of the smallest number and the largest number will be
divisible by 2 three times if the middle number is odd. Thus, if n is odd, the product (n – 1)(n + 1) must be
divisible by 2 three times.
(Consider why: If the middle number of 3 consecutive integers is odd, then the smallest and largest numbers
of the set will be consecutive even integers - their product must therefore be divisible by 2 at least twice.
Further, since the smallest and the largest number are consecutive even integers, one of them must be divisible
by 4. Thus the product of the smallest and largest number must actually be divisible by 2 at least three times!)
Additionally, given 3 consecutive positive integers, exactly ONE of those three numbers must be divisible by
3. To ensure that the product of the smallest number and the largest number will be divisible by 3, the middle
number must NOT be divisible by 3. Thus, for the expression n2 – 1 to be divisible by 24, n must be odd and
must NOT be divisible by 3.
Statement (2) alone tells us that n > 191. Since, this does not tell us whether n is even, odd, or divisible by 3, it
is not sufficient to answer the question.
Taking statements (1) and (2) together, we know that n is a prime number greater than 191. Every prime
number greater than 3 must, by definition, be ODD (since the only even prime number is 2), and must, by
definition, NOT be divisible by 3 (otherwise it would not be prime!)
Thus, so long as n is a prime number greater than 3, the expression n2 – 1 will always be divisible by 24. The
correct answer is C: Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but
NEITHER statement ALONE is sufficient.