Question

The largest natural number by which the product of three consecutive even natural numbers is always divisible is _______.
A.16
B.24
C.48
D.96

Answer 1

Hint: This is a problem of divisibility of numbers. This requires no formula, just the analysis of the expression and finding the number. A number is divisible by 2 if it is even.

Complete step-by-step answer:

Let the first number be 2n. We assumed the number to be 2n because it is an even number. Hence, the subsequent two even natural numbers are 2n + 2, and 2n + 4.
The numbers are-
2n, 2(n + 1) and 2(n + 2)
The product of these numbers is-
8n(n + 1)(n + 2)
We can clearly see that this number is divisible by 8. Also, at least one of the three n, n+1 and n+2 is an even number, and is divisible by 2. In three consecutive natural numbers, exactly one of the three numbers is divisible by 3. For example, 7, 8, 9 or 11, 12, 13.

Hence, the given number is divisible by 8, 2 and 3.
The largest natural number by which the product of three consecutive even natural numbers is always divisible is 8x2x3 = 48.

Hence, the correct option is C. 48.

Note: Students may write the answer as A. 16 because they do not consider the possibility of divisibility by 3, which is incorrect. Hence, it is necessary to check for every possibility.