This is a folk puzzle I found in one of my Class X textbooks under the Chapter - 'Real Numbers'. This has to be solved somehow using the concepts of Euclid's Division Lemma and LCM.
The puzzle goes as follows:
A trader was moving along a road selling eggs. An idler who didn't have much work to do started to get the trader into a wordy duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader asked the idler to pay for the eggs. When the idler asked how many eggs there were, the trader replied:
$$\text{If counted in pairs, one will remain;}$$ $$\text{If counted in threes, two will remain;}$$ $$\text{If counted in fours, three will remain;}$$ $$\text{If counted in fives, four will remain;}$$ $$\text{If counted in sixes, five will remain;}$$ $$\text{If counted in sevens, nothing will remain;}$$ $$\text{My basket cannot accommodate more than 150 eggs.}$$
How many eggs were there?
I could make the equations using the lemma:
$$\begin{align} a&=7p+0 \\ a&=6q+5 \\ a&=5r+4 \\ a&=4s+3 \\ a&=3t+2 \\ a&=2u+1 \end{align}$$
What do I do now?
Notice that $a+1$ is a multiple of $6,5,4,3,2$, hence is a multiple of their least common multiple, which is $60$.
Since $a$ is at most $150$, the possible values of $a$ are $59$ or $119$.
But then it's given that $a$ is a multiple of $7$.