Solving a Number Theory Problem: Finding Primes for a Divisibility Condition

🔍We can rewrite the given divisibility condition as a congruence statement using modular arithmetic.

🔢Euler's theorem states that if the gcd of a number a and a number n is equal to 1, then a to the power of Euler's totient function of n is congruent to 1 modulo n.

📊The order of an integer modulo n is the smallest positive exponent such that the power of the integer is congruent to 1 modulo n.

🧩By calculating the orders of 7 and 6 modulo 43, we find that the possible values for P are divisors of 42.

✅After analyzing the possible values of P, we conclude that the only solution to the given divisibility condition is P = 3.

What is Euler's theorem?

—Euler's theorem states that if the gcd of a number a and a number n is equal to 1, then a to the power of Euler's totient function of n is congruent to 1 modulo n.

What does the order of an integer modulo n mean?

—The order of an integer modulo n is the smallest positive exponent such that the power of the integer is congruent to 1 modulo n.

How did you calculate the orders of 7 and 6 modulo 43?

—We calculated the powers of 7 and 6 modulo 43 and observed the patterns to determine their orders. We found that the order of 7 is 6, and the order of 6 is 3 modulo 43.

What are the possible values for P in the given divisibility condition?

—The possible values for P are divisors of 42, which include 1, 2, 3, 6, 7, 14, 21, and 42.

What is the solution to the given divisibility condition?

—The only solution to the given divisibility condition is when P is equal to 3.

00:00In this video, we explore a number theory problem that involves finding primes for a specific divisibility condition.

03:08We apply Euler's theorem and analyze the orders of 7 and 6 modulo 43.

05:04We find that the possible values for P are divisors of 42.

06:33After analyzing the possible values of P, we conclude that the only solution to the given divisibility condition is P = 3.