What is x + y = ? x+y=?

Number Theory Level 2

If 10 ! 10! is divisible by 3 x 5 y 3^x5^y .

What is the largest value of x + y = ? x+y=?

Notation : ! ! is a factorial notation, for example 3 ! = 3 × 2 × 1 3!=3\times2\times1


The answer is 6.

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Since 10 ! = 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 = 2 8 ⋅ 3 4 ⋅ 5 2 ⋅ 7 10! = 10\cdot 9 \cdot\ 8 \cdot 7\cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 = 2^8\cdot3^{\color{#3D99F6}{4}}\cdot5^{\color{#D61F06}{2}} \cdot 7 , largest value of x + y {\color{#3D99F6}x} +{\color{#D61F06}y} is 4 + 2 = 6 {\color{#3D99F6} 4} + {\color{#D61F06} 2} = 6

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Thank you for sharing your solution.

Hana Wehbi - 3 years, 4 months ago

You missed 7 in 2^8 *3^4 *5^2.

Vaibhav Priyadarshi - 3 years ago

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fixed! thanks c:

Adolfo Henriquez Llanos - 3 years ago
Edwin Gray
Apr 10, 2019

Factors including 3 are 3,6,9, so there are 4 appearances of 3 Likewise for 5, the factors including 5 are 5,10, so there are 2 appearances of 5. 4 + 2 = 6.

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