Hi everyone

I am having with this question:

n is an integer.

Show that n^{5}- n is divisible by 30 for all integers n.

If it helps, in the questions before this one, we proved that the last digit of n^{5 }is n.

Thanks

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- June 7th 2012, 04:20 AMAndroidMachinePower to the 5
Hi everyone

I am having with this question:

n is an integer.

Show that n^{5}- n is divisible by 30 for all integers n.

If it helps, in the questions before this one, we proved that the last digit of n^{5 }is n.

Thanks - June 7th 2012, 06:21 AMemakarovRe: Power to the 5
So, the last digit of 786

^{5}is 786, right? (Smile)

One way is to factor n^{5}- n into 4 factors and then consider 5 cases depending on the remainder of n when divided by 5. In each case, find the factors among the 4 ones that are divisible by 2, 3 and 5.

Another way is to use mathematical induction. - June 7th 2012, 06:27 AMSorobanRe: Power to the 5
Hello, AndroidMachine!

I'll get you started . . .

Quote:

is an integer.

Show that is divisible by 30 for all integers

We must show that is divisible by 2, 3 and 5.

Factor: .

. . . . . . . . . . . .

. . . . . . . . . . . .

Facts: .The product of two consecutive integers is divisible by 2.

n . . . . The product of three consecutive integers is divisible by 3.

Now you must show that one of the four factors will be divisible by 5.