Final answer:
To prove that the expression 38⁹–38⁸ is divisible by 37, we can use the concept of modular arithmetic. By demonstrating that the expression is congruent to zero modulo 37, we can show that it is divisible by 37.
Step-by-step explanation:
To prove that the expression 38⁹–38⁸ is divisible by 37, we can use the concept of modular arithmetic. We need to show that the expression is congruent to zero modulo 37.
Let's write the expression as (38⁹) - (38⁸). We can use the property of modular arithmetic that states if a ≡ b (mod n), then aˣ ≡ bˣ (mod n).
In this case, we have 38 ≡ 1 (mod 37). Therefore, 38^9 ≡ 1^9 ≡ 1 (mod 37). Similarly, 38⁸ ≡ 1⁸ ≡ 1 (mod 37).
Substituting these values back into the expression, we get (38⁹) - (38⁸) ≡ 1 - 1 ≡ 0 (mod 37). This means that the expression is divisible by 37.