Consider a positive integer represented by the product: $2^4d^3e^5$ , where d and e are distinct prime numbers that are both greater than 2. How many distinct even factors does this integer possess?

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120

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Official Explanation

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To find the number of distinct even factors of the integer represented by

2^4d^3e^5

, we first note that the total number of factors of a number in the form

p_1^{e_1}p_2^{e_2}...p_n^{e_n}

is given by

(e_1 + 1)(e_2 + 1)...(e_n + 1)

. Here, the exponents are 4 for 2, 3 for d, and 5 for e. Thus, the total number of factors is

(4 + 1)(3 + 1)(5 + 1) = 5 \times 4 \times 6 = 120

. To find the number of even factors, we must…

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What this question tests

Your application of distance = rate × time to trains, boats, and commuters, including relative speed.

Mixing units (miles per hour vs. minutes) or averaging speeds — average speed is total distance over total time.