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Number Theory Level pending

For an odd number $n>1,$ what is the largest positive integer that $(n-1)(n+1)(n+3)$ is certainly divisible by?

The answer is 48.

1 solution

Alex Li
Mar 13, 2017

Since the number is odd, the number is the product of 3 consecutive even numbers multiplied together.

This implies that the number is divisible by 2^3. Furthermore, since every 2 even numbers are divisible by 2, it implies that the number is divisible by another 2. Also notice that there must be a multiple of 3, so the number is further divisible by 3.

This prime factorization gives us 2^4*3 = 48.

We can confirm that this is the minimum with the fact that, for n=3, 2 4 6 = 48.

Try writing the problem differently

The answer is 48.

1 solution

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