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Trailing zeros or largest power of 10 dividing a product or factorial: count 2s and 5s

This type of question, asked 7 times: CDS 1 2018, CDS 1 2019, CDS 2 2019, CDS 2 2021, CDS 1 2023, CDS 1 2025, CDS 1 2026. All 7 questions below, oldest first.

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  1. Question 1

    CDS 1 2018

    83
    What is the largest power of 10 that divides the product 1×2×3×4…×23×24×251 \times 2 \times 3 \times 4 \ldots \times 23 \times 24 \times 25 ?

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  2. Question 2

    CDS 1 2019

    86
    If 10n10^n divides 623×759×10526^{23} \times 75^9 \times 105^2, then what is the largest value of nn ?

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  3. Question 3

    CDS 2 2019

    2
    What is the largest value of nn such that 10n10^n divides the product 25×33×48×53×67×76×812×99×106×1512×2014×2211×25152^5 \times 3^3 \times 4^8 \times 5^3 \times 6^7 \times 7^6 \times 8^{12} \times 9^9 \times 10^6 \times 15^{12} \times 20^{14} \times 22^{11} \times 25^{15} ?

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  4. Question 4

    CDS 2 2021

    17
    How many zeros are there in the product 150×249×348×…×5011^{50}\times 2^{49}\times 3^{48}\times \ldots \times 50^{1} ?

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  5. Question 5

    CDS 1 2023

    2
    What is the largest power of 10 that divides the product 29×28×27×…2×129 \times 28 \times 27 \times \ldots 2 \times 1?

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  6. Question 6

    CDS 1 2025

    37
    Given that 100×99×98×…×3×2×1100n\frac{100 \times 99 \times 98 \times \ldots \times 3 \times 2 \times 1}{100^n} is an integer. What is the largest value of n for which this is true ?

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  7. Question 7

    CDS 1 2026

    5
    Let P=55×1515×2525×3535P = 5^5 \times 15^{15} \times 25^{25} \times 35^{35} and Q=1010×2020×3030×4040Q = 10^{10} \times 20^{20} \times 30^{30} \times 40^{40}. What is the number of consecutive zeros at the end of the sum (P+Q)(P + Q) ?

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