N is the number of buttons in a sewing box.

a). N is more than 40 but less than 80.

b).When N is divided by 5, the remainder is 2.

c).When N is divided by 7, the remainder is 4.

Find the value of N.

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- Dec 14th 2009, 05:57 PM #1

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- Dec 14th 2009, 06:37 PM #2
So you have $\displaystyle 40 < n < 80$.

$\displaystyle \Rightarrow$ The set of numbers is {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79}.

When $\displaystyle n$ is divided by $\displaystyle 5$, the remainder is two, so $\displaystyle n \equiv 2 \pmod{5}$. Thus $\displaystyle n = 2 + 5k$ with $\displaystyle k \in Z$.

$\displaystyle \Rightarrow$ The set of numbers is {2, 7, 12, ... 47, 52, 57, 62, 67, 72, 77, ...}

When $\displaystyle n$ is divided by $\displaystyle 7$, the remainder is four, so $\displaystyle n \equiv 4 \pmod{7}$. Thus $\displaystyle n = 4 + 7k$ with $\displaystyle k \in Z$.

$\displaystyle \Rightarrow$ The set of numbers is {4, 11, 18, ... 46, 53, 60, 67, 74, ...}

Now intersect those three sets : which numbers are in all three of them ?

Our three sets are :

{41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79}

{2, 7, 12, ... 47, 52, 57, 62, 67, 72, 77, ...}

{4, 11, 18, ... 46, 53, 60, 67, 74, ...}

In red, the numbers that are in the first set.

And in blue, the number that is in all three sets.

$\displaystyle \rightarrow$ Thus, $\displaystyle n = 67$.

- Dec 14th 2009, 06:57 PM #3