If $5^7$ is divided by $52$, then the remainder obtained is
Step-by-Step Solution
Key Concept: Use the binomial theorem to express powers in terms of a modulus and find the remainder by expanding $(a+b)^n$ where $a$ is divisible by the modulus.
We know that $5^1 = 625 = 52 \times 12 + 1$, so $5^1 = 52\lambda + 1$, where $\lambda$ is a positive integer. Then $(5^1)^{24} = (52\lambda + 1)^{24} = {}^{24}C_0(52\lambda)^{24} + {}^{24}C_1(52\lambda)^{23} + {}^{24}C_2(52\lambda)^{22} + \cdots + {}^{24}C_{23}(52\lambda) + {}^{24}C_{24} + 1 = 5^{24} = 52[{}^{24}C_0(52\lambda)^{23} + {}^{24}C_1(52\lambda)^{22} + \cdots + {}^{24}C_{23}\lambda] + 1 = (\text{a multiple of } 52) + 1$. On multiplying both sides by 5, we get the required result.
Correct Answer: 5