2023 ACSBR EM S1 EOY P2 Q4

Written as a product of its prime factors, \({648 = 2^3 \times 3^4}\). The number \({648k}\) is both a perfect square and a perfect cube. Find the smallest positive integer value of \({k}\).[1]

\({x}\) is a number between \({50}\) and \({100}\). The highest common factor of \({x}\) and \({84}\) is \({6}\). Find the smallest possible value of \({x}\).[2]

Two toy cars race around a circuit of equal sized tracks. They take \({12}\) seconds and \({21}\) seconds respectively to complete one round of the track. Given that the cars leave the starting point on each track at the same time, how long will it take, in minutes, before they are at the starting point together again?[2]