The expression a⁷- a is divisible by 7 if a is an integer.
Given that,
7 divides a^7−a
To find that if a is an integer, then 7 divides a^7 -a or a⁷- a is divisible by the number 7
We can write, where k is an integer
a^7−a=7k
a(a^6−1)=7k
Now if a=k then
a^6−1=7
a^6=8
and a=±1.414 so a^7−a=9.898
If a is not equal to k then
a(a^6−1)=a(a^3+1)(a^3−1)=7k
since a(a+1)(a−1) is the not product of 3 consecutive integers, the expression is divisible by 7.
Hence, The expression a⁷- a is divisible by 7 if a is an integer.
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