Question

The greatest common divisor of two numbers is 17 and their least common multiple is 765. How many pairs of values can the numbers assume?
(a)1
(b)2
(c)3
(d)4

Answer 1

Hint: The least number which is exactly divisible by each of the given numbers is called the least common multiple and the largest number that divides two or more numbers is the highest common factor (HCF). By using this definition, we calculate the required thing.

Complete step-by-step answer:
Factors and Multiples: All the numbers that divide a number completely, i.e., without leaving any remainder, are called factors of that number. Multiples are those numbers which we get after multiplying numbers.
The product with the highest power of the prime numbers that appear in prime factorization of any of the numbers gives us the LCM.
To find the HCF of the given numbers, we express each number as a product of prime numbers. The highest prime factor is HCF.
Now, proceeding to our question, since the HCF of numbers is 17. Hence, let the two numbers be 17x and 17y by using the above stated definition of HCF.
Also, we know that $\text{product of two numbers = }\left( \text{L}\text{.C}\text{.M} \right)\left( \text{H}\text{.C}\text{.F} \right)$.
From the details given in question,
$\begin{align}
  & 17x\times 17y=765 \\
 & xy=\dfrac{17\times 17\times 45}{17\times 17} \\
 & xy=45 \\
\end{align}$
Therefore, the possible pairs of x and y are: x = 1, y = 45 and x = 3, y = 15 and x = 5, y = 9. But their G.C.D is 17 so, the pair x = 3, y = 15 is cancelled.
Therefore, possible numbers are:
$\begin{align}
  & 17\times 1=17\text{ and }17\times 45=765 \\
 & 17\times 5=85\text{ and }17\times 9=153 \\
\end{align}$
Hence, two pairs are possible.
Therefore, option (b) is correct.

Note: The key step in solving this problem is the basic definition of LCM and HCF. Students must be careful while writing the possible pairs of multiples. They must remember that the pair should only include co-prime numbers so that G.C.D is 17.