the least number that is when successively divided by 7,5,3 leaves successive reminders 4,4,1 respectively is

Question

Options

172,144,102,109

my answer

is this the correct way to find the least number ?

Answer 1

No. You need

$$ 7(5(3+1)+4)+4=172$$

Work backwards. The smallest number that is $1\pmod3$ is $4$. $4$ is therefore the result of dividing by $5$, and therefore was $24$ before the 'divide by $5$ to leave remainder $4$' operation.

As the next operation is 'divide by $7$ to leave remainder $4$', then $24\times7=168$, and add $4$ to give $172$.