Question

Докажите,чтоа) 8 в 7 степени + 4 в 9 степени делится на 9б) 9 в 7 степени - 27 в 4 степени делится на 24в) 2 в 23 степени +4 в 11 степени - 8 в 7 степени делится на 40

Answer 1

а)

8 в 7 степени=2097152

4 в 9 степени=262144

(2097512+262144)/9=2359296/9=262144-делится

б)

9 в 7 степени=4782969

27 в 4 степени=531441

(4782969-531441)/24=4251528/24=177147-делится

в)

2 в 23 степени=8388608

4 в 11 степени=4194304

8 в 7 степени=2097152

(8388608+4194304)-2097152/40=(12582912-2097152)/40=10485760/40=262144-делится

Answer 2

Proof for a)

To prove that 8 raised to the power of 7 plus 4 raised to the power of 9 is divisible by 9, we can use modular arithmetic.

Let's calculate the remainder when 8 raised to the power of 7 plus 4 raised to the power of 9 is divided by 9:

8^7 + 4^9 ≡ (8^3)^2 * 8 + (4^3)^3 ≡ (512)^2 * 8 + (64)^3 ≡ 4^2 * 8 + 4^3 ≡ 16 * 8 + 64 ≡ 128 + 64 ≡ 192 ≡ 0 (mod 9)

Since the remainder is 0 when divided by 9, we can conclude that 8 raised to the power of 7 plus 4 raised to the power of 9 is divisible by 9.