Which one of the following is the remainder when 10^20 is divided by 7?
1
2
4
6
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CAPF – 2018
– First, simplify the base: 10 mod 7 = 3. So, 10 ≡ 3 (mod 7).
– Then, 10^20 ≡ 3^20 (mod 7).
– Now, we compute powers of 3 modulo 7 to find a pattern (cycle):
– 3^1 ≡ 3 (mod 7)
– 3^2 ≡ 9 ≡ 2 (mod 7)
– 3^3 ≡ 3 * 2 = 6 (mod 7)
– 3^4 ≡ 3 * 6 = 18 ≡ 4 (mod 7)
– 3^5 ≡ 3 * 4 = 12 ≡ 5 (mod 7)
– 3^6 ≡ 3 * 5 = 15 ≡ 1 (mod 7)
– The powers of 3 modulo 7 repeat with a cycle length of 6 (3, 2, 6, 4, 5, 1).
– To find 3^20 mod 7, we need to find the position in the cycle corresponding to the exponent 20. This is done by finding the remainder of 20 when divided by the cycle length 6.
– 20 ÷ 6 = 3 with a remainder of 2. (20 = 3 * 6 + 2).
– So, 3^20 ≡ 3^(6*3 + 2) ≡ (3^6)^3 * 3^2 (mod 7).
– Since 3^6 ≡ 1 (mod 7), we have:
– 3^20 ≡ (1)^3 * 3^2 (mod 7)
– 3^20 ≡ 1 * 9 (mod 7)
– 3^20 ≡ 9 ≡ 2 (mod 7).
– The remainder is 2.