Question

find a pair of natural numbers whose least common multiple is 78 and the greatest common divisor is 13​

Answer 1

Step-by-step explanation:

In this question, we are given that the least common multiple (LCM) of two numbers is 78 and the greatest common divisor (GCD) is 13.

Let these two numbers be x and y.

Now, we have a property that the product of two natural numbers x and y is equal to the product of its LCM and GCD. Therefore, we can say that for two natural numbers x and y,

⇒x×y=GCD(x,y)×LCM(x,y)

Here, we have GCQ equal to 13 and the LCM as 78. Therefore, substituting these values in above equation, we get

⇒x×y=13×78⇒x×y=1014

Now, on prime factorization 1014, we get

⇒x×y=2×3×13×13

Now, 13 is common divisor of both numbers x and y, we can write

⇒x×y=(2×13)×(3×13)

OR

⇒x×y=(1×13)×(2×3×13)

Hence, now x will be equal to (2×13) or (1×13) and y will be equal to (3×13) or (2×3×13). Therefore,

⇒x=2×13=26 OR ⇒x=1×13=13

⇒y=3×13=39 OR ⇒y=2×3×13=78

Hence, the two natural numbers whose LCM is 78 and GCD is 13 are 26 and 39 or 13 and 78

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