Question

Given that 198 = 2 x 3² x 11 and 90 = 2 x 3² x 5,

a) find the smallest integer, k, such that 198k is a perfect square

b) the largest integer that is a factor of both 198 and 90

(pls show working ty :) )

Answer 1

9514 1404 393

Answer:

a) 22

b) 18

Explanation:

a) In order for 198k to be a perfect square, all of its integer factors must be squares. 3² is already a factor. To make the other factors be squares, need to multiply by 2 and 11. That is, k = 2×11 = 22. Then we will have ...

198k = 2×3²×11 × (2×11) = 2²×3²×11² = (2×3×11)²

k = 22

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b) The factors that are different in the two numbers are 11 and 5. The factors that are the same are 2×3² = 18.

18 is the greatest common factor of 198 and 90

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The "work" is looking at the numbers in the factor lists and comparing those to what is needed to answer the question.