Consider the following statements 1 n 3 is divisible by 6 where n gt 3 2 n 3 3 n...........

Mathematics (NDA) Number System

Consider the following statements :

1. n!/3! is divisible by 6, where n > 3

2. n!/3! + 3 is divisible by 7, where n > 3

Which of the above statements is/are correct?

This questions was previously asked in

2022 NDA -1 Math

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Let’s break it down:

- Statement 1: n!/3! is divisible by 6, where n > 3

• n!/3! = n(n-1)(n-2)(n-3)...(4)

• For n > 3, this is a product of consecutive integers from 4 up, so it always contains a multiple of 2 and 3—so it’s divisible by 6 for any n > 3.

• The condition says n > 32, but for n > 3 this already holds, so statement 1 is correct.

- Statement 2: n!/3! + 3 is divisible by 7, for n > 3

• No clear pattern or theorem guarantees n!/3! + 3 is always divisible by 7 for n > 3.

• Try n = 4: (4! / 6) + 3 = (24/6) + 3 = 4 + 3 = 7 (divisible by 7)

• Try n = 5: (120/6) + 3 = 20 + 3 = 23 (not divisible by 7)

• So, statement 2 is not always correct.

? Option: 1 only is correct.

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