For each of the following pairs, verify the relationship given below:
LCM x HCF = Product of the given numbers
1) 9 and 11
2) 40 and 45
3) 20 and 28
4) 30 and 75
Answer (2)
To verify the relationship LCM × HCF = Product of the given numbers , we must:
Calculate the Least Common Multiple (LCM) and the Highest Common Factor (HCF) for each pair of numbers.
Multiply the LCM and HCF and check if it equals the product of the original numbers.
Let's go through each pair:
Pair: 9 and 11
HCF: 9 and 11 are both prime to each other, so HCF = 1 .
LCM: The LCM of two numbers when one is prime is the product of the two numbers, i.e., LCM = 9 × 11 = 99 .
Verification: 1 × 99 = 9 × 11 = 99 . The relationship holds.
Pair: 40 and 45
HCF: The factors of 40 are 1 , 2 , 4 , 5 , 8 , 10 , 20 , and 40 , and the factors of 45 are 1 , 3 , 5 , 9 , 15 , and 45 . The greatest common factor is 5, so HCF = 5 .
LCM: The multiples of 40 and 45 include their least common multiple, which is 360 .
Verification: 5 × 360 = 40 × 45 = 1800 . The relationship holds.
Pair: 20 and 28
HCF: The factors of 20 are 1 , 2 , 4 , 5 , 10 , and 20 , and the factors of 28 are 1 , 2 , 4 , 7 , 14 , and 28 . The greatest common factor is 4, so HCF = 4 .
LCM: The LCM of 20 and 28 is 140 .
Verification: 4 × 140 = 20 × 28 = 560 . The relationship holds.
Pair: 30 and 75
HCF: The factors of 30 are 1 , 2 , 3 , 5 , 6 , 10 , 15 , and 30 , and the factors of 75 are 1 , 3 , 5 , 15 , 25 , and 75 . The greatest common factor is 15, so HCF = 15 .
LCM: The LCM of 30 and 75 is 150 .
Verification: 15 × 150 = 30 × 75 = 2250 . The relationship holds.
In conclusion, for each pair of numbers given, the relationship LCM × HCF = the product of the numbers is verified and true.
By calculating the LCM and HCF for each pair of numbers, we confirmed that the equation LCM × HCF = Product of the numbers holds true for all pairs: 9 and 11, 40 and 45, 20 and 28, and 30 and 75.
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