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 Question from Richie, a parent: Hi there, There are 2 parts to the question. #1. Factorise 90. This is easy, 90=2x3x3x5 #2 LCM of 6,15 and x is 90. What are possible values of x if x is odd? From #1, since 90=6x15, how can this be used to work out possible values of x? The answers given for x are 9 and 45. Thanks in advance, Richie

Hi Richie,

Since the LCM of $6, 15$ and $x$ is $90 = 2 \times 3^2 \times 5$ the prime decomposition of $x$ cannot contain any primes other than $2, 3$ and $5.$ Since x is odd it cannot have $2$ as a prime factor Also since $90 = 2 \times 3^2 \times 5,$ the prime decomposition of $x$ can contain at most two threes and at most one five. The LCM of $6 = 2 \times 3$ and $15 = 3 \times 5$ is $2 \times 3 \times 5 = 30$ and since the LCM of $6, 15$ and $x$ is $90, x$ must contribute another $3.$ Thus $x$ could be $3 \times 3 = 9$ or $3 \times 3 \times 5 = 45.$

Penny

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