How many eggs she had? Is there more than one possibility to solve it?

Question

She put the eggs in group of two,there was one egg left over. When she put them in groups of three, there was also one egg left over.The same thing happened when she put them in groups of four.But when she puts them in groups of five, she ended up with complete groups of five with no egg left over.

Answer 1

When N items are put into groups of n and r items are left over, it means that the total number of items N must be r more than some multiple of n.

N = nq + r

where

N: total number of items

n: items per group

q: number of groups

r: number left over, the remainder

For example, when 17 is put into groups of 5, there will be 3 groups with 2 left over.

17 = 5(3) + 2

here

N: 17

n: 5

q: 3

r: 2

In general, if we don't specify how many groups (q) of 5 we can form, it will still be true that any number N in the form N = 5q + 2 will have 2 left over.

22 = 5(4) + 2

22 put into groups of 5 forms 4 groups with 2 left over

27 = 5(5) + 2

27 put into groups of 5 forms 5 groups with 2 left over

32 = 5(6) + 2

32 put into groups of 5 forms 6 groups with 2 left over

N = 5q + 2

N put into groups of 5 forms q groups with 2 left over

For this question, the number of eggs E is put into groups of different numbers and the number left over reveals that it must be possible to express the total number in the following ways:

E = 2q + 1

In groups of 2, there is 1 left over

E = 3q + 1

In groups of 3, there is 1 left over

E = 4q + 1

The same is true for groups of 4

E = 5q + 0

In groups of 5, no eggs are left over

By comparing the expressions, it's possible to see that the smallest positive number that fits all these forms is 25.

25 = 2(12) + 1

In groups of 2, there are 12 groups with 1 left over

25 = 3(8) + 1

In groups of 3, there are 8 groups with 1 left over

25 = 4(6) + 1

In groups of 4, there are 6 groups with 1 left over

25 = 5(5) + 0

In groups of 5, there are 5 groups with no eggs left over

There could be 25 eggs.

Note: It's possible to show that any number in the form 25 + 60n (ex. 25, 85, 145, 205...) could also be the number of eggs.

Answer 2

Since a group of 5 goes evenly into a number ending in 0 or 5. But it cannot end in 0 or it would be divisible by 2. So, you know that the number of eggs ends with a 5!

Start with trial and error.....choose

5 NO

15 NO

25 YES 25/3 = 8 with remainder 1, 25/4 = 6 with remainder 1, 25/5 = 5 no remainder