If 653xy is divisible by 80, find x + y

Given data

• Number: 653xy
• Divisibility requirement: by 80

Concept / Approach

• 80 = 16 × 5 (coprime). So the number must be divisible by 5 and by 16.
• Divisible by 5 → last digit y = 0.
• Divisible by 16 → last four digits are divisible by 16.

Step-by-step calculation

With y = 0, last four digits are 53x0 = 5300 + 10x.Compute (5300 + 10x) mod 16.5300 mod 16 = 4 (since 16 × 331 = 5296)We need 4 + 10x ≡ 0 (mod 16).Reduce: 10x ≡ 12 (mod 16) ⇒ divide by 2 ⇒ 5x ≡ 6 (mod 8).Inverse of 5 mod 8 is 5 (since 5 × 5 ≡ 1 mod 8).x ≡ 6 × 5 ≡ 30 ≡ 6 (mod 8). As a digit, x = 6.Thus y = 0 and x + y = 6.

Verification

Last four digits 5360 ÷ 16 = 335 (exact); last digit 0 satisfies divisibility by 5.

Common pitfalls

• Testing divisibility by 8 instead of 16; 80 requires 16 × 5, not 8 × 10.

Final Answer

x + y = 6.