A number theory problem by Nikhil Raj

Number Theory Level 1

Find sum of digits of ∏ n = 5 9 n ∑ a = 1 20 a \displaystyle{\huge \dfrac{\prod_{n=5}^{9}n}{\sum_{a=1}^{20}a}}


The answer is 9.

Nikhil Raj by Nikhil Raj , 19, India

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1 solution

Nikhil Raj
May 31, 2017

We have, ∏ n = 5 9 n ∑ a = 1 20 = 5 × 6 × 7 × 8 × 9 1 + 2 + … + 20 = 5 × 6 × 7 × 8 × 9 20 2 ( 1 + 20 ) = 5 × 6 × 7 × 8 × 9 10 × 21 = 72 ∴ Sum of digits = 7 + 2 = 9 \dfrac{\prod_{n=5}^9n}{\sum_{a=1}^{20}} = \dfrac{5 \times 6 \times 7 \times 8 \times 9}{1 + 2 + \ldots + 20} = \dfrac{5 \times 6 \times 7 \times 8 \times 9}{\frac{20}{2}(1 +20)} = \dfrac{5 \times 6 \times 7 \times 8 \times 9}{10 \times 21} = 72 \\ \therefore {\text{Sum of digits}} = 7 + 2 = \boxed{9}

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