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I'm asked to find the number of elements of order $24$ in $C_8 \times C_8 \times C_3 \times C_5$. From the solutions, I can sort of see how to do this. However, on a similar question I'm asked to find the number of elements of order $196$ in $C_4 \times C_4 \times C_{49} \times C_7$, and the method doesn't look the same. We never really went over this in lectures but it's in all the exam papers, what is the general method to solve these?

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    Hi @Jam, it is always a good idea tell us what have you tried so far. – 2017-01-03
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    Note the $C_24$ and $C_196$ are subgroups of the two groups respectively. – 2017-01-03

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The basic idea is that the order of an element $(w,x,y,z)$ in the group $C_{n_1}\times C_{n_2}\times C_{n_3}\times C_{n_4}$ with coprime $n_i$ is the least common multiple of the orders of $w$,$x$,$y$ and $z$ in the corresponding cyclic group $C_{n_i}$. This reduces the question to the number of elements of order $d$ in a cyclic group $C_n$. Now the number of elements of order $d$, where $d$ is a divisor of $n$, is $\varphi(d)$. There are many examples given on MSE (including what to do for prime powers) see for example this duplicate, concerning the number of elements of order $196$ in $C_4 \times C_4 \times C_{49} \times C_7$.