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# Help plz. Thx

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What is the smallest positive integer that is the sum of a multiple of $15$ and a multiple of $21$? (Remember that multiples can be negative.)

Dec 24, 2023

#1
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To find the smallest positive integer that is the sum of a multiple of $15$ and a multiple of $21$, we need to find the least common multiple (LCM) of $15$ and $21$.

The prime factorization of $15$ is $3 \times 5$, and the prime factorization of $21$ is $3 \times 7$. The LCM is the product of the highest powers of all prime factors involved:

LCM(15,21)=31×51×71=105.LCM(15,21)=31×51×71=105.

Therefore, the smallest positive integer that is the sum of a multiple of $15$ and a multiple of $21$ is $105$.

To confirm, we can check that $105$ is a multiple of both $15$ and $21$:

105=7×15=5×21.105=7×15=5×21.

So, $105$ is indeed the smallest positive integer satisfying the given conditions.

Dec 24, 2023
#2
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What is the smallest positive integer that is the sum of a multiple of $15$ and a multiple of $21$? (Remember that multiples can be negative.)

(15)(3) + (21)(–2)

45 – 42  =  3

You can get it another way.

(21)(3) + (15)(–4)

63 – 60  =  3

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Dec 25, 2023
#3
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(21)(1/7) + (15)(–1/5)

3 – 3  =  0

Nowhere did it say that the multiple had to be an integer.

The only problem I see with this is the question, is zero positive?

.

Dec 29, 2023
edited by Bosco  Dec 29, 2023