If 42573* is divisible by 72, find the minimum value of

Given data

• Number: 42573
• Divisibility requirement: by 72

Concept / Approach

• Since 72 = 8 × 9 and gcd(8, 9) = 1, the number must be divisible by 8 and by 9 simultaneously.

Step-by-step calculation: divisibility by 8

Only the last three digits matter: 73*Nearest multiples of 8 around 730 are 728 and 736 and 744736 fits the pattern 73* ⇒ * = 6

Step-by-step calculation: divisibility by 9

Sum of digits (without *) = 4 + 2 + 5 + 7 + 3 = 21We need (21 + *) to be a multiple of 9 ⇒ * ≡ 6 (mod 9)Smallest digit satisfying both tests is 6

Verification

425736 ÷ 8 = 53217; 4 + 2 + 5 + 7 + 3 + 6 = 27 divisible by 9.

Common pitfalls

• Checking only one of the two conditions; 72 requires both 8 and 9.

Final Answer

Minimum value of * is 6.