Question

Which of the following
numbers having its fourth
power ending with 1?
*​

Answer 1

Answer:

Any number A can be written as:

A = 10knk + 10k−1nk−1 + ... + 10n1 + n0,

where nk is the starting digit of the number and can take on any integer from 0 to 9 and so forth.

If we raise this to the fourth power and don't combine any of the terms, each term should be divisible by 10 except maybe not n40. Now, n40 must end must end in k where k satisfies:

n40 ≡k(mod10).

And since n40 ends in k, A must end in k (since all of the other terms in the sum are divisible by 10). So, it suffices to just check what the last digit of the fourth powers of 1 through 9 are (since 0≤n0≤9):

04 ≡0(mod10)

14≡1(mod10)

24≡6(mod10)

34≡1(mod10)

44≡6(mod10)

54≡5(mod10)

64≡6(mod10)

74≡1(mod10)

84≡6(mod10)

94≡1(mod10).

So the fourth power of any integer must end in either a 0,1,5, or 6.

Did I get lucky or is this ok? Are there other more elegant ways?

Thank you

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