Divisibility by 8 (last three digits rule) What least digit should replace * in 91876*2 so that the number is divisible by 8?

Introduction / Context:
A number is divisible by 8 if and only if its last three digits form a number divisible by 8. This problem asks you to choose the least digit replacing * to satisfy that condition.

Given Data / Assumptions:

• Number pattern: 91876*2.
• Only the last three digits affect divisibility by 8.
• We must find the smallest digit that works.

Concept / Approach:
Consider the last three-digit number formed: 6*2. Let * = d. Then the value is 600 + 10d + 2. Reduce this expression modulo 8 to enforce divisibility.

Step-by-Step Solution:
Compute (600 + 10d + 2) mod 8.Since 600 mod 8 = 0 and 10 mod 8 = 2, expression ≡ (0 + 2d + 2) mod 8.We need 2d + 2 ≡ 0 (mod 8) ⇒ 2d ≡ 6 (mod 8) ⇒ d ≡ 3 (mod 4).Digits satisfying this are 3 and 7; the least such digit is 3.

Verification / Alternative check:
Test d = 3: last three digits 632; 632 ÷ 8 = 79 exactly. Hence the full number is divisible by 8.

Why Other Options Are Wrong:
1, 2, 4 lead to last-three-digit numbers 612, 622, 642, which are not divisible by 8.

Common Pitfalls:
Checking more than the last three digits; forgetting to reduce 10 to 2 modulo 8; picking 7 (also valid) when the question asks for the least such digit.

Final Answer:
3