How many eggs?(Euclid's Division lemma)

[dweera]

Well i have come across this problem in one of the textbooks. The text book solves it by trial and error method. I don't understand how.
The problem goes :
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs. How many eggs were there?

 

[ksdhart2]

Okay, so let's carefully read the problem text and see what information we can gather from it. In this case, I think starting with the final statement is the best course of action.

My basket cannot accomodate more than 150 eggs.

This is pretty straightforward. We now know an upper limit. Common sense tells us that we can't have a zero eggs, a negative number of eggs, or a fractional number of eggs in the basket. But that leaves 150 possible solutions. Let's make a list of the numbers between 1 and 150, keep reading and see if we can't eliminate some of those solutions.

If counted in pairs, one will remain

What do you think it means to "count [the eggs] in pairs?" What does it mean that when you do so, there's one egg left over? Which numbers can you cross off your list based on this information?

If counted in threes, two will remain

What do you think it means to "count [the eggs] in threes?" What does it mean that when you do so, there's two eggs left over? Which numbers can you cross off your list based on this information?

Keep going through each of the restrictions, crossing off numbers as you go along. What are you left with at the end? Are you left with one solution? Are you left with multiple solutions?