2017 AMC 8 Problems/Problem 12

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Problem

The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?

$\textbf{(A) }2\text{ and }19\qquad\textbf{(B) }20\text{ and }39\qquad\textbf{(C) }40\text{ and }59\qquad\textbf{(D) }60\text{ and }79\qquad\textbf{(E) }80\text{ and }124$

Solution 1

Since the remainder is the same for all numbers, then we will only need to find the lowest common multiple of the three given numbers, and add the given remainder. The $\operatorname{LCM}(4,5,6)$ is $60$. Since $60+1=61$, that is in the range of $\boxed{\textbf{(D)}\ \text{60 and 79}}.$

Solution 2

Call the number we want to find $n$. We can say that $$n \equiv 1 \mod 4$$ $$n \equiv 1 \mod 5$$ $$n \equiv 1 \mod 6.$$ We can also say that $n-1$ is divisible by $4,5$, and $6.$ Therefore, $n-1=lcm(4,5,6)=60$, so $n=60+1=61$ which is in the range of $\boxed{\textbf{(D)}\ \text{60 and 79}}.$

~PEKKA

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