2024 · Q99
Let S = 5ᵃ+7ᵇ+11ᶜ+13ᵈ, where a, b c and d are natural numbers. What is the number of distinct remainders of S when it is divided by 10?
Elementary MathematicsNumber System
- A1
- B4
- C5
- DMore than 5
Explanation
5^a ≡ 5, 11^c ≡ 1 (mod 10); 7^b cycles 7,9,3,1 and 13^d ≡ 3^d cycles 3,9,7,1. So S ≡ 6 + 7^b + 3^d (mod 10), and 7^b+3^d takes the 5 distinct residues {0,2,4,6,8}. Hence S has 5 distinct remainders — option C.