Find the remainder when 67^67 + 67 is divided by 68.

Given data

• Compute (67^67 + 67) mod 68.
• Notation note: The original '(6767 + 67)' is treated as 67^67 + 67, a standard remainder problem; this preserves values and intent.

Concept / Approach

• Work modulo 68 using the relation 67 \u2261 −1 (mod 68).

Step-by-step calculation

67 \u2261 −1 (mod 68) \u2192 67^67 \u2261 (−1)^67 = −1 (mod 68)Thus, 67^67 + 67 \u2261 (−1) + (−1) = −2 \u2261 66 (mod 68)

Verification

Because 68 divides 67 + 1, any odd power gives −1; adding another 67 (also −1) yields −2 \u2192 remainder 66.

Common pitfalls

• Reducing mod 67 instead of mod 68; the key is 67 \u2261 −1 (mod 68).
• Treating (67^67 + 67) as 67^(67+67).

Final Answer

66