The number 247_679 is completely divisible by 9. What is the smallest whole digit that can replace the blank so that the resulting number is divisible by 9?

Introduction / Context:
This question checks understanding of the divisibility rule for 9. The rule is that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9. A missing digit is represented by a blank, and we must find the smallest digit that makes the whole number divisible by 9. Such problems are very common in quantitative aptitude and basic number theory tests.

Given Data / Assumptions:

• The number has the form 247d679, where d is an unknown digit from 0 to 9.
• The full number must be divisible by 9.
• We must choose the smallest whole digit d that makes this happen.

Concept / Approach:
The divisibility rule for 9 says that the sum of all digits must be a multiple of 9. So we compute the sum of all known digits, express the total sum as that value plus d, and then find the smallest digit d between 0 and 9 that makes the total sum a multiple of 9. This is much faster than trying to divide each candidate number by 9 directly.

Step-by-Step Solution:
Write the number as 2, 4, 7, d, 6, 7, 9.Sum of known digits = 2 + 4 + 7 + 6 + 7 + 9.Compute this sum: 2 + 4 = 6, 6 + 7 = 13, 13 + 6 = 19, 19 + 7 = 26, 26 + 9 = 35.So the total digit sum is 35 + d.For divisibility by 9, 35 + d must be a multiple of 9.The multiples of 9 above 35 are 36, 45, 54, and so on.Check 36 first because we want the smallest possible d.If 35 + d = 36, then d = 36 - 35 = 1.Digit d must be between 0 and 9, and 1 satisfies this.Therefore the smallest whole digit that works is 1.

Verification / Alternative check:
If the number is 2471679 (with d = 1), recompute the sum of digits.Digit sum = 2 + 4 + 7 + 1 + 6 + 7 + 9 = 36.Since 36 is a multiple of 9, the number 2471679 is divisible by 9.If we tried any smaller digit such as 0, the sum would be 35, which is not divisible by 9, so 1 is indeed the smallest value.

Why Other Options Are Wrong:
If d = 2, the digit sum is 37, not a multiple of 9.If d = 5, the sum is 40, and if d = 7, the sum is 42. None of these are multiples of 9.Hence 2, 5, and 7 do not make the number divisible by 9.

Common Pitfalls:
Some learners may confuse the rule for 9 with the rule for 3, but both use the digit sum; the difference is in the multiple used.Others may try to work with the full number and do long division, which is slower and more error prone than using the digit sum method.A common mistake is to accept the first valid d without ensuring it is the smallest, but in this case 1 is already the smallest possible valid digit.

Final Answer:
The smallest digit that can replace the blank is 1.