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Find the pairs of natural numbers whose greatest common divisor is 5 and the least common
multiple is 105.

seo-qna
Last updated date: 22nd Mar 2024
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Views today: 4.89k
MVSAT 2024
Answer
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Hint: Use the fact that GCD$ \times $LCM = 525 = the product of each of the required pairs of
natural numbers.
Factorize 525 in pairs. Consider the pairs with 5 as a divisor and find their GCD and LCM.
The pairs that satisfy the given condition would be the required answers.

Complete step by step solution:
The information given to us is:
The greatest common divisor (GCD) of a pair of natural numbers is 5.
And the least common multiple (LCM) of the same pair should be 105.
Let us use the following fact:
The product of a pair of natural numbers = GCD of the those natural numbers$ \times $LCM of those
natural numbers
So, we have GCD$ \times $LCM$ = 5 \times 105 = 525$
Thus, the required pair of numbers should have their product as 525.
Consider all the pairwise factorizations of 525.
They are:
\[
525 = 3 \times 175 \\
525 = 5 \times 105 \\
525 = 7 \times 75 \\
525 = 15 \times 35 \\
\]
We need those natural numbers with GCD as 5. This would mean they should have at least 5 as their
common divisor.
Therefore, we can rule out \[525 = 3 \times 175\]and\[525 = 7 \times 75\]as 3 and 175 do not have 5 as
their common divisor nor do 7 and 75.
Consider the pair of 5 and 105.

Here, the GCD is 5 because 5 cannot be divided by any number greater than itself.
And the LCM of these two numbers is definitely 105 as 105 is the smallest multiple of itself and it is also
a multiple of 5.
So, this means one of the required pairs is (5, 105).
Now consider the pair of 15 and 35.
15 can be written as$15 = 5 \times 3$
Similarly, 35 can be written as$35 = 5 \times 7$
Both 3 and 7 are prime numbers.
Therefore, the greatest common divisor of 15 and 35 is 5.
Now, consider the multiples of 15 and 35.
The multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120 etc.
The multiples of 35, 70, 105 etc.
As can be seen, the least common multiple of 15 and 35 is 105.
Thus, the second and the only other pair of natural numbers is (15, 35).
Hence the required pairs are (5, 105) and (15, 35).

Note: We should remember that GCD of a pair of numbers is the greatest common divisor of itself and
the LCM. Similarly, LCM is the least common multiple of the GCD and LCM.
Therefore, in such questions, the pair of GCD and LCM will always be one of the answers.