word problem - elementary number theory - divisibility

Question

Show that the product of three consecutive integers is divisible by 504 if the middle one is a cube

Answer 1

We need $$a^3(a^6-1)$$ divisible by $504=7\cdot2^3\cdot3^2$

Now by Fermat's Little Theorem $7$ divides $a^7-a$ which divides $a^3(a^6-1)=a^2(a^7-a)$

Now if $2|a,$ $$2^3|a^3$$ hence $a^3(a^6-1)$

else $2\nmid a, (2,a)=1$ using Carmichael Function, $$2^3|(a^{2^{3-1}}-1)=a^2-1$$ for odd $a$

Check for $3^2$