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    Question

    Determine the remainder when 4412 is divided

    by 7?
    A 5 Correct Answer Incorrect Answer
    B 3 Correct Answer Incorrect Answer
    C 1 Correct Answer Incorrect Answer
    D 4 Correct Answer Incorrect Answer

    Solution

    4412 ÷ 7 = (42 + 2)12

    By using binomial theorem, we have

    (42 + 2)12 = 12C0 × 4212 × 20 + 12C1 × 4211 × 21 + …………+ 12C12 × 420 × 212

    So, in the above expression only the last term will not be completely divisible by ‘7’.

    So, required remainder = Remainder obtained when 12C12 × 420 × 212 is divided by ‘7’ or 212 is divided by ‘7’.

    Now, 212 ÷ 7 = (23)4 ÷ 7 = 84 ÷ 7 = (7 + 1)4 ÷ 7

    So, again by binomial expansion, we get

    (7 + 1)4 = 4C0 × 712 × 10 + 4C1 × 711 × 11 + …………+ 4C4 × 70 × 14

    So, last term will not be divisible by ‘7’.

    So, required remainder = 4C4 × 70 × 14 = 1

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