# 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?

##### 2 Answers

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:

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

And so by observation it's 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

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

Check:

Algebraically,

This is where the multiple comes in. To get 40 < n < 70 we multiply it by 2.

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