Question

A woman with a basket of epos finds that if she removes the eggs from the basket 3 of 7 at a time, there is always 1 009 kt. However, it she removes the eggs 8 at a time, there are no eggs let the basket holds up to 100 eggs, how many eggs does shehave?

Answer 1

We hava a basket of eggs, but we don't know how many there are in the basket.

Lets call X the number of eggs in the basket.

We know that if we take 3 eggs at a time or 7 eggs at a time, there are one egg left in the basket.

That means the residual of the division X/3 and X/7 is 1.

We know that if we take 8 eggs at a time, there are no eggs left in the basket.

That means the residual of the division X/8 is 0 (or X is multiple of 8).

We also know X is less or equal to 100.

`\begin{gathered} X-3m=1 \\ X-7n=1 \\ X-8h=0 \end{gathered}`

We have to find a number that is divisible by 7 and 3, and then add 1 and verify that is divisible by 8.

As 7 and 3 have no common denominator, we have to find a number that is divisible by 7*3=21 and then add 1 and verify that is divisible by 8.

We can check one by one:

If we add 1 to 21, we have 22. It is not divisible by 8, so it is not the value of X.

The next number is (2*21) + 1 = 42 + 1 = 43. It is not divisible by 8.

The next number is (3*21) + 1 = 63 + 1 = 64. It is divisible by 8, so this is the value of X.

The number of eggs in the basket is 64.

Equations

X-21n=1

X-8m=0

8m=1-21n

m=(1-21n)/8 and m has to be a positive integer.

We can iterate n=1,2,3.... until we find an m that is a positive integer.