The 5-digit number 7a35b is divisible by 45, where a and b are positive integers. What is the value of (a + b)?
12
Divisibility by 45 requires divisibility by both 9 and 5.
Divisibility by 5: the last digit b must be 0 or 5. Since b is a positive integer, b = 5.
Divisibility by 9: the digit sum must be divisible by 9. Digit sum \(= 7 + a + 3 + 5 + b = 7 + a + 3 + 5 + 5 = 20 + a\).
For divisibility by 9: \(20 + a \equiv 0 \pmod{9}\), i.e., \(20 + a\) is a multiple of 9.
The next multiple of 9 after 20 is 27, so \(a = 27 - 20 = 7\).
Check: a = 7 is a positive single digit, so the number is 77355 = 77355. Digit sum = 27 ✓, last digit 5 ✓.
Therefore, \(a + b = 7 + 5 = 12\).
Hence, the value of (a + b) is 12.
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