**Number Puzzle #81**

A box contains beer bottles if divided between 4 friends 4 bottles are balance and if divided between 5 friends 3 bottles are balance what will be the balance if divided between 7 friends the box contains minimum no of bottles.

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**Answer:** There will be one extra bottle after dividing between 7 friends.

**How?**

Let’s say total number of beer bottles in box are x. then we can say

**x = 4*m + 4**, where m is integer greater than or equal to 0.

**x = 5*n + 3**, where n is integer greater than or equal to 0.

From above two eequations

4m+4 = 5n + 3 = x

=> 5n – 4m = 1, where m and n are integers greater or equal to 0.

For bottles to be minimum, lets assume m and n are 0

thus from above equation 0 = 1, which is not true, so they cant be both 0 at same time.

If you observe neither of them can be zero at a time, so Now lets take next case where m and n are 1.

5m -4n = 1

5 – 4 = 1

1 = 1, which is true

thus total number of bottles are 5+3 or 4 + 4, i.e. 8

and if divided between 7 friends, 1 bottle will be extra.