Categories

# Understand the problem

Suppose that for a prime number $p$ and integers $a,b,c$ the following holds:
$$6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.$$
Prove that $p\mid a,b,c$.

Baltic Way 2009

Number Theory
Medium
##### Suggested Book
Elementary Number Theory by David Burton

Do you really need a hint? Try it first!

Show that, $a^4+b^4+c^4\equiv 2(b^2+bc+c^2)^2$ modulo $p$. Hence $p| (b^2+bc+c^2)$. From there, try to find a relation between $b$ and $c$ modulo $p$.
As $b^2+bc+c^2|b^3-c^3$, we have $b^3\equiv c^3\;\text{mod}\;p$. Note that, $\frac{p+1}{3}$ is an integer and hence $b^{p+1}\equiv c^{p+1}$. Using Fermat’s little theorem, conclude that $b^2\equiv c^2$. Use this along with $b^3\equiv c^3$ to conclude that either $b\equiv c$ or $b,c$ are both divisible by $p$.
Clearly, if $p|b,c$ then we are done.  Prove that the case $b\equiv c$ can be reduced to this one.
If $b\equiv c$, then we have $a\equiv -2b$ and $a^4\equiv -2b^4$. These together give $p|18b^4$. Using $6|p+1$, conclude that $p|b$. The proof is now concluded using hint 3.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Triangle Problem | PRMO-2018 | Problem No-24

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## What is Parity in Mathematics ? 🧐

Parity in Mathematics is a term which we use to express if a given integer is even or odd. It is basically depend on the remainder when we divide a number by 2. Parity can be divided into two categories – 1. Even Parity 2. Odd Parity Even Parity : If we...

## Value of Sum | PRMO – 2018 | Question 16

Try this Integer Problem from Number theory from PRMO 2018, Question 16 You may use sequential hints to solve the problem.

## Chessboard Problem | PRMO-2018 | Problem No-26

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## Measure of Angle | PRMO-2018 | Problem No-29

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

## Good numbers Problem | PRMO-2018 | Question 22

Try this good numbers Problem from Number theory from PRMO 2018, Question 22 You may use sequential hints to solve the problem.

## Polynomial Problem | PRMO-2018 | Question 30

Try this Integer Problem from Number theory from PRMO 2018, Question 30 You may use sequential hints to solve the problem.

## Digits Problem | PRMO – 2018 | Question 19

Try this Integer Problem from Number theory from PRMO 2018, Question 19 You may use sequential hints to solve the problem.

## Chocolates Problem | PRMO – 2018 | Problem No. – 28

Try this beautiful Problem on Combinatorics from PRMO -2018.You may use sequential hints to solve the problem.

## Trigonometry | PRMO-2018 | Problem No-14

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

This site uses Akismet to reduce spam. Learn how your comment data is processed.