Question

Let p>5 be the prime number. Prove that the expression p^4-10p^2+9 is divisible by 1920

Answer 1

Answer:

By the division algorithm, we can write each p as

p=10q+r

for some quotient q and remainder 0≤r<10. As p is prime, clearly r≠0. r also cannot be even, otherwise p is even. Finally, note that r≠5 or 5∣p.

For a more brief and algebraic solution, note that (p, 10)=1 implies that p is a unit in Z/10Z. The units of the ring are precisely 1, 3, 7 and 9.

For completeness, you should probably show that there exists primes for each of the remaining congruence classes.