If 653xy is divisible by 90, find x + y.

Given data

• Five-digit number: 6,5,3,x,y; divisible by 90.

Concept / Approach

• 90 = 9 × 10; a number divisible by 90 must be divisible by both 9 and 10.
• Divisible by 10 ⇒ last digit y = 0.
• Divisible by 9 ⇒ sum of digits multiple of 9.

Step-by-step calculation

Digit sum = 6 + 5 + 3 + x + y = 14 + x + yWith y = 0 ⇒ 14 + x ≡ 0 (mod 9) ⇒ x ≡ −14 ≡ 4 (mod 9)Valid digit x = 4; hence x + y = 4 + 0 = 4

Verification

65340 ÷ 90 = 726 exactly; digit sum 6+5+3+4+0=18 (multiple of 9) and ends with 0.

Common pitfalls

• Forgetting the divisibility-by-10 condition (y must be 0).

Final Answer

4