Question

How many zeros are at the end of 459 · 885? Explain how you can answer this question without actually computing the number. (Hint: 10 = 2 · 5.) When this number is written in ordinary decimal form, each 0 at its end comes from a factor of , or one factor of 2 and one factor of . Since there are factors of 2 and factors of 5, there are exactly factors of 10 in the number. This implies that the number ends with zeroes.

Answer 1

To find the number of trailing zeros in the product of 459 and 885, we look for factors of 2 and 5, which together form a factor of 10. Since both numbers end in non-zero digits, there are no pairs of these factors, hence no trailing zeros in the product.

Step-by-step explanation:

To determine how many zeros are at the end of the product of 459 · 885 without actual computation, we analyze the number of 2s and 5s in their prime factorization, as each zero in the final product is created from one pair of these factors (a factor of 10). However, since both numbers end with a non-zero digit, we can tell that there are no factors of 10 present, and thus the product does not end with any zeros.

Answer 2

Step-by-step explanation:

If the multiplication of two numbers has the zero at the end, one of them will have a factor of 2 and the other number will have the factor of 5.

Example: Multiplication of 25 and 8.

Factors of 25 = 5 × 5 [5 is a factor of 25]

Factors of 8 = 2 × 2 × 2 [2 is a factor of 8]

Multiplication of 8 × 25 = 200 [Zero at the end]

Two numbers are 459 and 885.

Factors of 459 = 3 × 3 × 3 × 17

Factors of 885 = 3 × 5 × 59

[Since, 2 is not a factor in both the numbers multiplication of the numbers will not have 0 at the end]

459 × 885 = 406215