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Let us find the remainder when (77^77 + 77) is divided by 78.

Step 1: Express in terms of modulus.
We need to find (77^77 + 77) mod 78.

Step 2: Note that 77 = 78 - 1, so 77 ≡ -1 (mod 78).

Step 3: Substitute:
77^77 ≡ (-1)^77 ≡ -1 (mod 78)

Step 4: So,
77^77 + 77 ≡ -1 + 77 ≡ 76 (mod 78)

Step 5: Therefore, the remainder is 76.

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