the product of two numbers is 1521 and the hcf of these numbers is 13. find the number of such pairs?
the product of two numbers is 1521 and the hcf of these numbers is 13. find the number of such pairs?
Answer: To find the number of pairs of numbers whose product is 1521 and whose highest common factor (HCF) is 13, we can use the following approach:
Step 1: Prime Factorization
First, let’s find the prime factorization of 1521.
1521 = 3^2 * 13^2
Step 2: Determining Possible Factors
Since the HCF of the numbers is 13, both numbers must have at least one 13 as a factor. The remaining factors are distributed between the two numbers such that their product becomes 1521.
We have three factors to allocate: 3^2 and 13^2. Let’s consider the possible combinations:
Combination 1:
Number 1: 13^1 * 3^0
Number 2: 13^1 * 3^2 = 507
Combination 2:
Number 1: 13^0 * 3^2 = 9
Number 2: 13^2 * 3^0 = 2197
Combination 3:
Number 1: 13^2 * 3^0 = 169
Number 2: 13^0 * 3^2 = 9
Step 3: Checking Validity
Now, let’s check if these pairs satisfy the condition that their HCF is 13 and their product is 1521.
Combination 1:
HCF(13, 507) = 13
Product = 13 * 507 = 6591 ≠ 1521
Combination 2:
HCF(9, 2197) = 1 ≠ 13
Product = 9 * 2197 = 19773 ≠ 1521
Combination 3:
HCF(169, 9) = 1 ≠ 13
Product = 169 * 9 = 1521 (satisfies the condition)
Step 4: Conclusion
There is only one pair of numbers (169, 9) whose product is 1521 and HCF is 13. Therefore, there is only one such pair that meets the given criteria.