If 1 0 ! is divisible by 3 x 5 y .
What is the largest value of x + y = ?
Notation: ! is a factorial notation, for example 3 ! = 3 × 2 × 1
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Yes, exactly how we solve it. Thank you for sharing a nice solution.
No prob, Hana! Thanks for your many problem posts here :)
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I enjoy it.
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Me too.......Officially 3 years & counting with Brilliant! Are you doing the 100 Days of Summer Challenge?
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@Tom Engelsman – Yes. I like this kind of questions.
@Hana Nakkache , it would simpler to ask for: "What is the largest value of x + y = ? "
Can anyone help me with the relevant wiki for my solution. I don't remember the name of the theorem, but it's a great one.
I got bored at home so I did this. it's interesting until I did it.
Relevant Wiki: Legendre's formula
x , which is the exponent for 3, can be obtained by:
⌊ 3 1 1 0 ⌋ + ⌊ 3 2 1 0 ⌋ + < 1 … = 4
Similarly, y can also be found out:
⌊ 5 1 1 0 ⌋ + < 1 … = 2
Thus, the sum is:
x + y = 6
@Mahdi Raza , Thank you for sharing your solution, nice approach to solve it too. Isn't this the wiki you are looking for: https://brilliant.org/wiki/floor-function/
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Welcome. Not exactly that wiki, it's related but I was looking for the wiki which relates factorials and primes with a floor function. I can't remember it... I will let you know which one if I find it. Thanks!
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I found it: Legendre's formula
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Taking 1 0 ! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 ⋅ 8 ⋅ 9 ⋅ 1 0 = 1 ⋅ 2 1 ⋅ 3 1 ⋅ 2 2 ⋅ 5 1 ⋅ 2 1 3 1 ⋅ 7 1 ⋅ 2 3 ⋅ 3 2 ⋅ 2 1 5 1 = 2 8 3 4 5 2 7 1 , then 3 x 5 y ⇒ x = 4 , y = 2 and x + y = 6 .