Divisibility by 9 (digit-sum rule) What least digit should replace * in 6135*2 so that the number is exactly divisible by 9?

Introduction / Context:
Divisibility by 9 is determined by the sum of digits: the digit sum must be a multiple of 9. Here we choose the smallest digit * making 6135*2 divisible by 9.

Given Data / Assumptions:

• Digits fixed except * in 6135*2.
• Digit sum with * included must be 9k.
• We want the least digit that works.

Concept / Approach:
Compute the current digit sum and then find * so that the total is the next multiple of 9. This is faster than trial division.

Step-by-Step Solution:
Digit sum without *: 6+1+3+5+2 = 17.We need 17 + * to be a multiple of 9.The next multiple of 9 after 17 is 18, so * = 1.Therefore, the least suitable digit is 1.

Verification / Alternative check:
With * = 1, full sum is 18; numbers with digit sum 18 are divisible by 9. Quick test by dividing confirms exact divisibility.

Why Other Options Are Wrong:
0 gives sum 17; 2 gives 19; 3 gives 20. None are multiples of 9.

Common Pitfalls:
Choosing 10 − (17 mod 9) = 1 by mistake for 10-based rounding rather than 9; forgetting that the target must be a multiple of 9.

Final Answer:
1