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1. How many elements of order 8 are there in [itex]C_{45}[/itex]?

Solution: [itex]\varphi(8)=4[/itex]

2. How many elements of order 2 are there in [itex]C_{20}\times C_{30}[/itex]?

Solution: [itex]C_{20}\times C_{30}\simeq C_2\times C_4\times C_3\times C_5\times C_5[/itex]

Since 2 is coprime to 3 and 5 the question is equivalent to how many elements are there in [itex]C_2\times C_4[/itex]? Answer is [itex]\varphi(2)\varphi(4)=3[/itex]

3. How many elements of order 35 are there in [itex]C_{100}\times C_{42}[/itex]?

Solution: Since 2,4 and3 are coprime to 35 the question is equivalent to how many elements of order 35 are there in [itex]C_{25}\times C_7?[/itex]

Answer is [itex]\varphi(25)\varphi(7)=120[/itex]

Now this sounds a bit large to me. A peer of mine said the answer is [itex]\varphi(7)\varphi(5)=24[/itex]. Can anyone explain the reasoning behind this?