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.)

hellomeeee Dec 24, 2023

#1**-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.

BuiIderBoi Dec 24, 2023