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:

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 xx 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:
62/5 = 12 + 2 and 62/6 = 10 + 2

Algebraically,
(n-2)/5 = A and (n-2)/6 = B

(n-2) = 5A = 6B ; A = 6/5B

A/B = 6/5 ; A = 6 , B = 5

5A = (n-2) ; 5 xx 6 = n - 2 ; 30 + 2 = n
6B = (n-2) ; 6 xx 5 = n - 2 ; 30 + 2 = n

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

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