Is this statement congruent and why?
Is 83 to 144 power = 15 to the 144 power (mod 17)
it's true, since 83 = 15 (mod 17)
To determine whether the statement is congruent, we need to compare the given exponentiation of two numbers modulo 17.
Let's break down the problem into steps to find our answer:
Step 1: Evaluate the given exponentiation expressions.
- Calculate 83 to the power of 144 (mod 17).
- Calculate 15 to the power of 144 (mod 17).
Step 2: Compare the results from step 1.
- If the two results are the same, then the statement is congruent.
- If the two results are NOT the same, then the statement is NOT congruent.
Now, let's evaluate each expression and find out whether they are congruent.
Step 1: Evaluate the given exponentiation expressions.
To calculate 83 to the power of 144 (mod 17), we need to take the remainder after dividing 83^144 by 17. However, calculating such a large number would be impractical, so we can use a modular property to simplify the calculation:
- Property: (a * b) mod n ≡ ((a mod n) * (b mod n)) mod n
Using this property, we can divide the exponentiation into smaller steps:
Step 1.1: Calculate 83 (mod 17)
- 83 mod 17 = 15
Step 1.2: Calculate 15^144 (mod 17)
- As 15 and 17 are relatively prime, we can use Euler's totient theorem: a^(φ(m)) ≡ 1 (mod m), where φ(m) is the totient function.
- The totient function of 17 is φ(17) = 17 - 1 = 16.
- So, 15^144 (mod 17) ≡ 15^(144 mod 16) (mod 17) ≡ 15^0 (mod 17) ≡ 1 (mod 17)
Thus, 83^144 (mod 17) ≡ 1 (mod 17)
Now let's calculate 15^144 (mod 17).
Step 1.3: Calculate 15^144 (mod 17)
- To use the modular property, we divide the exponentiation into steps.
- First, we calculate 15 (mod 17) = 15.
Step 1.4: Calculate 15^144 (mod 17)
- Similar to Step 1.2, we can use Euler's totient theorem: 15^(144 mod 16) (mod 17).
- 144 mod 16 = 0, so we get 15^0 (mod 17) ≡ 1 (mod 17).
Thus, 15^144 (mod 17) ≡ 1 (mod 17)
Step 2: Compare the results.
The first exponentiation expression is congruent to 1 (mod 17), and the second exponentiation expression is also congruent to 1 (mod 17). Since both results are the same, we can conclude that:
83 to the 144 power is congruent to 15 to the 144 power (mod 17).
Therefore, the statement is congruent.