Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and a -= c (mod n), then a^3 -= c^3(mod n).

OlmekinjP 2021-01-04 Answered

Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n>1, and ac(mod n),then a3c3(mod n).

You can still ask an expert for help

Want to know more about Congruence?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

tafzijdeq
Answered 2021-01-05 Author has 92 answers

Step 1
Let a, c and n be integers such that n>1.
We have to prove that if ac(modn),then a3c3(modn) using the definition of congruence modulo n.
Note that, pq(modn)means n(pq) where p and q are integers.
So, by the definition of congruence modulo n, ac(modn)tn^(ac).
If n divides (ac), then n divides any integer multiple of (ac).
Step 2
Since a and c are integers, a2+ac+c2 is also an integer.
Now, n divides any integer multiple of (a−c) implies that n divides (ac)(a2+ac+c2).
But, by the algebraic identity (ac)(a2+ac+c2)=a3c3.
So, we can say that n divides (a3c3)andthus n(a3c3).
Therefore, by the definition of congruence modulo n, a3c3(modn).

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more