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What is the largest number k less than 1000 such that the remainder is 1 when k is divided by each of 3, 4, and 5?

 May 1, 2022
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961. lcm(3,4,5) = 60 so k will be in the form ----> k = 60n + 1    for some integer n

 

the problem asks for largest k less than 1000 so:       k < 1000 ----> 60n+1 < 1000

 

solving for n the inequality yields n < 17 and since we are looking for greatest int less than 1000 then n will equal 16 in this case to maximize k.

 

plugging in n = 16

 

k = 60(16) + 1 = 961

 May 1, 2022

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