# Thread: value of N

1. ## value of N

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.

2. So you have $40 < n < 80$.
$\Rightarrow$ 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 $n$ is divided by $5$, the remainder is two, so $n \equiv 2 \pmod{5}$. Thus $n = 2 + 5k$ with $k \in Z$.
$\Rightarrow$ The set of numbers is {2, 7, 12, ... 47, 52, 57, 62, 67, 72, 77, ...}

When $n$ is divided by $7$, the remainder is four, so $n \equiv 4 \pmod{7}$. Thus $n = 4 + 7k$ with $k \in Z$.
$\Rightarrow$ 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.
$\rightarrow$ Thus, $n = 67$.

3. thank u for ur help!