Question

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

Answer 1

Hint: We know that the product of two numbers is equal to the product of greatest common divisor of two numbers and least common multiple of two numbers. Let us assume two numbers a and b. Now we have to find the product of a and b.

Complete step-by-step answer:
Let us assume the product of a and b is equal to P. Let us assume this equation as equation (1). Let us assume the greatest common divisor of a and b is equal to C. Let us assume this equation as equation (2). Let us assume the least common multiple of a and b is equal to D. Let us assume this equation as equation (3). We know that the product of two numbers is equal to the product of greatest common divisor of two numbers and least common multiple of two numbers. From the question, we were given that the greatest common divisor is 5 and the least common multiple is 105. So, we have to find the product of 5 and 105. Now we have to find all the possibilities such that the product of two numbers whose product is equal to product of 5 and 105. Before solving the question, we know that the product of two numbers is equal to the product of greatest common divisor of two numbers and least common multiple of two numbers.
Let us assume two numbers a and b.
Now we have to find the product of a and b. Let us assume the product of a and b is equal to P.
\[\Rightarrow P=a\times b......(1)\]
Let us assume the greatest common divisor of a and b is equal to C.
\[\Rightarrow G.C.D=C....(2)\]
Let us assume the least common multiple of a and b is equal to D.
\[\Rightarrow L.C.M=D....(3)\]
We know that the product of two numbers is equal to the product of greatest common divisor of two numbers and least common multiple of two numbers.
So, from equation (1), equation (2) and equation (3), we get
\[\Rightarrow a\times b=C\times D.......(4)\]
From the question, it was given that the greatest common divisor of two numbers is equal to 5.
\[\begin{align}
  & \Rightarrow G.C.D=5 \\
 & \Rightarrow C=5......(5) \\
\end{align}\]
From the question, it is also clear that the least common multiple of two numbers is equal to 105.
\[\begin{align}
  & \Rightarrow L.C.M=105 \\
 & \Rightarrow D=105......(6) \\
\end{align}\]
Now we will substitute equation (5) and equation (6) in equation (4), we get
\[\begin{align}
  & \Rightarrow a\times b=5\times 105 \\
 & \Rightarrow a\times b=525......(7) \\
\end{align}\]
From equation (7), it is clear that the product of a and b is equal to 525.
Now we have to the set of pairs (a, b) such that the product of a and b is equal to 525.
\[\begin{align}
  & 525=1\times 525 \\
 & 525=3\times 175 \\
 & 525=5\times 105 \\
 & 525=15\times 35 \\
 & 525=21\times 25 \\
 & 525=25\times 21 \\
 & 525=35\times 15 \\
 & 525=105\times 5 \\
 & 525=175\times 3 \\
 & 525=525\times 1 \\
\end{align}\]
So, we can say that there are 10 possible pairs of (a, b) for which the product of a and b is equal to 525.

Note: Students should write the correct values of a and b such that the product of a and b is equal to 525. If any product is written wrong, the solution may get interrupted. So, students should have a clear view. Students should also be able to write all the possible ways such that the product of two numbers is equal to 525. If any possible product is not written, then the final answer gets interrupted. So, students should have a clear view.