Question

Доказать что число 333^777 + 777^333 делится на 37. Подробно.

Answer 1

$333^{777}=(3\cdot 111)^{777}=3^{777}\cdot 111^{777}\\ \\ 777^{333}=(7\cdot 111)^{333}=7^{333}\cdot 111^{333}$

Значит $333^{777}+777^{333}=3^{777}\cdot 111^{777}+7^{333}\cdot 111^{333}=111^{333}\cdot\left(3^{777}\cdot111^{444}+7^{333}\right)$

Разложим число 111 на простые множители

111 = 3 * 37

$111^{333}\cdot\left(3^{777}\cdot111^{444}+7^{333}\right)=3^{333}\cdot 37^{333}\cdot\left(3^{777}\cdot111^{444}+7^{333}\right)$

Множитель $37^{333}$ делится на 37, следовательно, сумма чисел $\left(333^{777}+777^{333}\right)$ делится на 37

Answer 2

To prove that the number 333^777 + 777^333 is divisible by 37, we need to show that the remainder when dividing this number by 37 is zero.

Let's proceed with the proof step by step:

Step 1: Determine the remainder when 333 is divided by 37. Step 2: Determine the remainder when 777 is divided by 37. Step 3: Use the properties of remainders to simplify 333^777 + 777^333. Step 4: Show that the simplified expression is divisible by 37.

Step 1: Remainder of 333 when divided by 37: 333 ÷ 37 = 9 with a remainder of 12.

Step 2: Remainder of 777 when divided by 37: 777 ÷ 37 = 21 with a remainder of 0.

Step 3: Simplify the expression 333^777 + 777^333 using the properties of remainders:

First, consider 333^777: Since the remainder of 333 when divided by 37 is 12, we can express 333 as 37n + 12, where n is an integer.

Now, 333^777 = (37n + 12)^777

When you expand this binomial, all the terms will have powers of 37n raised to some exponent. The constant term will be the one with the highest power of 37, and all other terms will have additional factors of 37. Therefore, all those terms except the constant term will be divisible by 37.

So, 333^777 ≡ 12^777 (mod 37)

Next, consider 777^333: Since the remainder of 777 when divided by 37 is 0, we can express 777 as 37m + 0, where m is an integer.

Now, 777^333 = (37m)^333

Since any number raised to the power of 333 is still divisible by 37, we can conclude that 777^333 is divisible by 37.

Step 4: Combining the simplified expressions:

333^777 + 777^333 ≡ 12^777 + 0 (mod 37)

Now, we need to check if 12^777 is divisible by 37.

12^1 ≡ 12 (mod 37) 12^2 ≡ 144 ≡ 33 (mod 37) 12^3 ≡ 12 * 33 ≡ 396 ≡ 10 (mod 37) 12^4 ≡ 120 (mod 37) 12^5 ≡ 12 * 120 ≡ 1440 ≡ 13 (mod 37)

The powers of 12 keep repeating in the sequence: 12, 33, 10, 120, 13, ...

Since 777 is divisible by 4 (777 = 4 * 194), the power 777 will be at the same position in the sequence as 12^1.

So, 12^777 ≡ 12 (mod 37)

Therefore, 333^777 + 777^333 ≡ 12 + 0 ≡ 12 (mod 37)

Since 12 is not divisible by 37, we can conclude that 333^777 + 777^333 is NOT divisible by 37.

Apologies, but it seems there was an error in my initial response. The number 333^777 + 777^333 is not divisible by 37.