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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
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 🙂 )
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Mathematics
5 years
2021-09-04T14:40:48+00:00
2021-09-04T14:40:48+00:00 1 Answers
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Answer:
a) 22
b) 18
Step-by-step 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.