Compute an exact division by 11 quickly: Evaluate 777777 ÷ 11 using an efficient mental strategy.

Introduction / Context:
Dividing a repeating-digit integer by 11 can often be done mentally. Recognizing patterns or verifying with quick multiplication helps avoid long division while ensuring accuracy.

Given Data / Assumptions:

• N = 777777.
• We require N ÷ 11.
• All operations are exact with no remainder expected.

Concept / Approach:
A direct approach is to test the most plausible quotient and confirm by multiplication. A helpful pattern: 70707 × 11 = 70707 × (10 + 1) = 707070 + 70707 = 777777, matching the dividend exactly. This confirms the quotient efficiently.

Step-by-Step Solution:
Assume candidate q = 70707.Compute q × 11 = q × 10 + q = 707070 + 70707 = 777777.Therefore, 777777 ÷ 11 = 70707.

Verification / Alternative check:
Perform standard long division or apply the 11-divisibility alternating-sum technique to reassure yourself the number is divisible by 11 and then back-multiply to confirm exactness.

Why Other Options Are Wrong:
7077, 7707, 7007 produce products with 11 that are far from 777777, revealing incorrect magnitude.

Common Pitfalls:
Mixing up the number of digits in the quotient; attempting quick mental division but dropping carries. The back-multiplication check is the safest guarantee.

Final Answer:
70707