A number leaves remainder 11 when divided by 13 and remainder 9 when divided by 17. Find the least such number.

Given data

• N ≡ 11 (mod 13)
• N ≡ 9 (mod 17)

Concept / Approach

• Chinese Remainder Theorem on mod 13 and mod 17 (coprime, so unique solution mod 221).

Step-by-step calculation

Let N = 11 + 13k.Impose mod 17: 11 + 13k ≡ 9 (mod 17) ⇒ 13k ≡ −2 ≡ 15 (mod 17).Since 13·4 ≡ 52 ≡ 1 (mod 17), multiply both sides by 4 ⇒ k ≡ 15·4 ≡ 60 ≡ 9 (mod 17).Smallest k = 9 ⇒ N = 11 + 13·9 = 128.General solution: N = 128 + 221t, t ∈ ℤ.

Verification

• 128 ÷ 13 = 9 remainder 11 ✓
• 128 ÷ 17 = 7 remainder 9 ✓

Final Answer

Least number = 128.