CSAT 2024 Solutions - How many consecutive zeros are there at the end of the integer obtained in the product 1^2 × 2^4 × 3^6 × ... × 25^50?

Q9. How many consecutive zeros are there at the end of the integer obtained in the product 1² × 2⁴ × 3⁶ × 4⁸ ×.... × 25⁵⁰?

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A. 50
B. 55
C. 100
D. 200

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Answer: D. 200

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Explanation:

To find the number of trailing zeros in the product 1² × 2⁴ × 3⁶ × ... × 25⁵⁰, we need to count the number of times 10 appears in this product. Since each 10 is created by a pair of 2 and 5, and there are always more factors of 2, the number of zeros at the end of the product depends on the number of factors of 5.

Steps:

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1. Identify terms that contribute factors of 5: We look at multiples of 5 in the sequence and count their factors of 5:

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- 5¹⁰ contributes 10 factors of 5

- 10²⁰ contributes 20 factors of 5

- 15³⁰ contributes 30 factors of 5

- 20⁴⁰ contributes 40 factors of 5

- 25⁵⁰ contributes 100 factors of 5

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2. Add up the factors of 5:

10 + 20 + 30 + 40 + 100 = 200

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So, the product has **200 trailing zeros**.

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