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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
avatar+222 
-1

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
avatar+676 
0

 

 

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  

.

 Dec 25, 2023
 #3
avatar+676 
0

 

 

                   (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

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