Question

If the two numbers are $35$ and $49$, find a third number such that the$\mathrm{LCM}$ of the three numbers is $1225$.

Answer 1

Determine the third number of LCM.

As we know that the Least Common Multiple $\left(\mathrm{LCM}\right)$ of two or more numbers like $a$ and $b$ is the smallest positive number that is divisible by both $a$ and $b$.

We can find the $\mathrm{LCM}$ of any number by using the following methods,

1. Listing Multiples method.
2. Prime Factorization method
3. Division Method

It is given that the $\mathrm{LCM}$ of three numbers are $1225$, and the two numbers are $35$ and $49$.

Let the third number be $x$.

We will find the $\mathrm{LCM}$ of given terms by using Prime Factorization method.

Prime Factorization of $35$ are,

$35=5\times 7$

The numbers $5$ and $7$ are all prime numbers, therefore no further Factorization is possible.

Prime Factorization of $49$ are,

$49=7\times 7$

The number $7$ is a prime number, therefore no further Factorization is possible.

Prime Factorization of $1225$ are,

$1225=5\times 245$

$245=5\times 5\times 49$

$49=5\times 5\times 7\times 7$

The numbers $5$ and $7$ are all prime numbers, therefore no further Factorization is possible.

Compute a number comprised of factors that appear in at least one of the following:

$35,49,1225=5\times 5\times 7\times 7$

Multiply the numbers,

$5\times 5\times 7\times 7=1225$

Since the $\mathrm{LCM}$ of the three numbers is $1225$, the third number should be $1225$.

Hence, $\mathrm{LCM}\left(35,49,1225\right)$ is $1225$.