Counting Zeros in a Product

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Questions and Answers

Match the following factors of 5 in the product 15 × 16 × 17 × 18 × 20 × 21 × 24 × 25 × 27 × 28 with their corresponding number of factors of 5:

15 = 1 20 = 1 25 = 2 18 = 0

Match the prime factors in the product 15 × 16 × 17 × 18 × 20 × 21 × 24 × 25 × 27 × 28 with their corresponding power in the product:

3 = 4 2 = 12 5 = 4 7 = 2

Match the following numbers with their number of factors of 2:

16 = 4 20 = 2 24 = 3 28 = 2

Match the following concepts related to counting trailing zeros in a product with their definitions:

<p>Trailing zeros = Zeros at the end of a number Factors of 5 = Numbers that divide a number evenly Factors of 10 = Numbers that are multiples of 10 Exponent of 5 = Determines the number of trailing zeros in a product</p> Signup and view all the answers

Match the following steps in calculating the number of trailing zeros in a product with their corresponding description:

<p>Find the factors of 5 = Count the number of times 5 appears in the product Find the factors of 2 = Count the number of times 2 appears in the product Determine the smaller count = Identify the limiting factor in the product Calculate the trailing zeros = The number of trailing zeros is equal to the smaller count of factors</p> Signup and view all the answers

Flashcards

Trailing Zeros

The number of zeros at the end of a product.

Factors of 10

10 is composed of 2 and 5 as factors.

Counting Factors of 5

Identify how many times 5 is a factor in the product.

Counting Factors of 2

Count the occurrences of 2 in the product's factors.

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Determining Trailing Zeros

The lesser of the counts of factors of 2 and 5 gives the number of trailing zeros.

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Study Notes

Counting Zeros in a Product

The given product is 15 × 16 × 17 × 18 × 20 × 21 × 24 × 25 × 27 × 28.
To count the number of zeros at the end of the product, determine the number of times 10 is a factor.
10 = 2 × 5. Thus, count the factors of 5 and 2 in the product. The exponent of 5 determines the number of trailing zeros.
The prime factorizations of the numbers in the product are:
- 15 = 3 × 5
- 16 = 2⁴
- 17 = 17
- 18 = 2 × 3²
- 20 = 2² × 5
- 21 = 3 × 7
- 24 = 2³ × 3
- 25 = 5²
- 27 = 3³
- 28 = 2² × 7
Counting factors of 5:
- 5 appears in 15, 20, and 25. This yields 1 + 1 + 2 = 4 factors of 5.
Counting factors of 2:
- 2 appears in 16, 18, 20, 24, and 28. This results in 4 + 1 + 2 + 3 + 2 = 12 factors of 2.
Since there are 4 factors of 5 and 12 factors of 2, the number of factors of 10 is limited by the smaller count, which is 4.
Therefore, there are 4 trailing zeros in the product.