Find the order of the given elements in the direct product. (i) ([8],[10]) ∈ Z12 × Z18 (ii) ([3],[6],[12],[16]) ∈ Z4 × Z12 × Z20 × Z24?

(i) 8 has order 3 in Z12, since 8 * 3 = 0 (mod 12), and no smaller positive integer n satisfies 8n = 0 (mod 12).

10 has order 9 in Z18, since 10 * 9 = 0 (mod 18), and no smaller positive integer n satisfies 10n = 0 (mod 18).

Hence, (8, 10) has order lcm(3, 9) = 9 in Z12 x Z18.
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(ii) Similarly,
3 has order 4 in Z4
6 has order 2 in Z12
12 has order 5 in Z20
16 has order 3 in Z24.

Thus, (3, 6, 9, 12) has order lcm(4, 2, 5, 3) = 60 in Z4 x Z12 x Z20 x Z24.
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I hope this helps!