To determine the value of [tex]\( A \)[/tex] that makes the number [tex]\( 74A567 \)[/tex] divisible by 9, we use the rule of divisibility for 9. This rule states that a number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Here are the steps to solve the problem:
1. Write down the digits of the given number, [tex]\( 74A567 \)[/tex].
2. Identify the known digits and the unknown digit [tex]\( A \)[/tex].
Known digits: [tex]\( 7, 4, 5, 6, 7 \)[/tex]
Unknown digit: [tex]\( A \)[/tex]
3. Calculate the sum of the known digits:
[tex]\[
7 + 4 + 5 + 6 + 7 = 29
\][/tex]
4. To maintain divisibility by 9, the sum of all the digits (including [tex]\( A \)[/tex]) should be a multiple of 9. Let’s denote the sum of the known digits as [tex]\( S \)[/tex]:
[tex]\[
S = 29
\][/tex]
5. We need to find [tex]\( A \)[/tex] such that [tex]\( S + A \)[/tex] is divisible by 9. This means:
[tex]\[
(S + A) \mod 9 = 0
\][/tex]
6. Substitute [tex]\( S = 29 \)[/tex] into the equation:
[tex]\[
(29 + A) \mod 9 = 0
\][/tex]
7. Calculate [tex]\( 29 \mod 9 \)[/tex] to determine the remainder when 29 is divided by 9:
[tex]\[
29 \div 9 = 3 \text{ remainder } 2
\][/tex]
Thus,
[tex]\[
29 \mod 9 = 2
\][/tex]
8. For the sum [tex]\( (29 + A) \)[/tex] to be divisible by 9, the value of [tex]\( (2 + A) \mod 9 \)[/tex] must equal 0. Therefore, [tex]\( A \)[/tex] must be such that:
[tex]\[
2 + A \equiv 0 \pmod{9}
\][/tex]
9. Solve for [tex]\( A \)[/tex]:
[tex]\[
A = 9 - 2 = 7
\][/tex]
Thus, the value of [tex]\( A \)[/tex] that makes the number [tex]\( 74A567 \)[/tex] divisible by 9 is:
[tex]\[
\boxed{7}
\][/tex]
