Find the least number which is a multiple of $31$ and when divided by $15, 24$ and $32$ leaves the remainders $2, 11$ and $19$,respectively.

The traffic lights at three different road crossings change after every $48\, seconds,$ $72\, seconds$ and $108\, seconds,$ respectively. If they all change simultaneously at $8: 20: 00$ $hours,$ then they will again change simultaneously at

The $LCM$ of $2^{2} \times 3^{2} \times 5 \times 11$,$2^{4} \times 3^{4} \times 5^{2} \times 7$,and $2^{5} \times 3^{3} \times 5^{2} \times 7^{2} \times 11$ is:

The least number which when divided by $5, 6, 7$ and $8$ leaves $3$ as a remainder,but when divided by $9$ leaves no remainder,is:

The $LCM$ of two numbers is $495$ and their $HCF$ is $5.$ If the sum of the numbers is $100,$ then their difference is

Find the maximum possible length which can be used to measure exactly the lengths $6 \, m$,$4 \, m \, 75 \, cm$,and $10 \, m \, 25 \, cm$. (in $, cm$)