# A number in the range of 40 to 70 can be divided by 5 and have a remainder of 2. It can also be divided by 6 and also have a remainder of 2. What's the number?

62

#### Explanation:

One way we can do this is to list out the possibilities:

For a number divided by 5 with remainder of 2, we'll have a number that ends in either 2 or 7:

$42 , 47 , 52 , 57 , 62 , 67$

For a number divided by 6 with remainder of 2, we'll have:

$44 , 50 , 56 , 62 , 68$

And so by observation it's 62.

Oct 15, 2017

$62$

#### Explanation:

To find the number, it must be the Least Common Multiple of both 5 and 6, times a multiplier, if necessary to get into the desired range, plus 2. Because 5 is prime, the LCM is simply $5 \times 6 = 30$

That is not in our desires 40 < n < 70 range. Doubling it puts it into the range, as 60. Tripling would again put it outside of the range.

Therefore, the number evenly-divisible by 5 and 6 in that range is 60. To have a remainder of 2 in either case means that is must be $60 + 2 = 62$

Check:
$\frac{62}{5} = 12 + 2$ and $\frac{62}{6} = 10 + 2$

Algebraically,
$\frac{n - 2}{5} = A$ and $\frac{n - 2}{6} = B$

$\left(n - 2\right) = 5 A = 6 B$ ; $A = \frac{6}{5} B$

$\frac{A}{B} = \frac{6}{5}$ ; $A = 6$ , $B = 5$

$5 A = \left(n - 2\right)$ ; $5 \times 6 = n - 2$ ; $30 + 2 = n$
$6 B = \left(n - 2\right)$ ; $6 \times 5 = n - 2$ ; $30 + 2 = n$

This is where the multiple comes in. To get 40 < n < 70 we multiply it by 2. $30 \times 2 = 60$. $n + 2 = 62$

That could also be done with an inequality equation, but this seems simpler.