Number System

[chilly basin]

Consider the following statements:

I. If n is a prime number greater than
5, n⁴-1 is divisible by 2400.
II. Every square number is of the form 5n or (5n-1) or (5n+1) where n is a
whole number.

Which of the statement(s) given is/are
correct?
(a) Only I
(b)Only II
(c)Both I and II
(d)Neither I nor II

[chilly basin]

Unbelievable. No one answered

[wheat kite]

damn

(n²+1)(n+1)(n-1) = 2400k

[chilly basin]

Oh really?

[chilly basin]

Please put n=11

[jolly skiff]

anything ending in a 1 squared will also end in a 1, likewise with 9,
2, 8 will always end in 4
3, 7 will always end in 9
4, 6 will always end in 6
5 will always end in 5

So it's safe to say using a bit of reasoning that any whole number squared will satisfy 5n, 5n-1 or 5n+1

[jolly skiff]

the first statement can be disproven by example like with n=11

so your answer is probably b

[chilly basin]

I solved that. My problem was the other statement

[chilly basin]

So, hit and trial is the only method to find answers for such questions?

What if we have to be sure (if the case was otherwise) that a number is divisible by 2400 (here) (just assume)

Can we prove it by principle of mathematical induction?

[jolly skiff]

it's not the only way by any means but it's certainly the quickest!

I'm sure it could totally be proved / disproved using induction yeah

[chilly basin]

Would be glad if someone can do it by induction