LCM of two numbers is 20 and their HCF is 5 product of two numbers is
Answers
Step-by-step explanation:
Answer: 20 and 4
Proof 1:
Let us call the two numbers as first and second number and break down the L.C.M. and the H.C.F. into their constituent prime factors.
L.C.M. = 20 = 2 x 2 x 5
H.C.F. = 4 = 2 x 2
From the very definition, H.C.F. is the greatest number that divides both the undetermined numbers. Therefore, 2x2 are factors of the two numbers. This gives us 4 as one of the two unknown numbers.
From definition of L.C.M., the two unknown numbers divide the L.C.M. This means 2x2x5 (= 20) are factors of the two numbers. Of the 3 factors 2,2, and 5, two (2,2) have already been counted. We are left only with 5. Therefore, the second number is 4x5 = 20
Thus, the two numbers are: 20 and 4 (Proved)
Correctness of the result is easily verifiable since, by inspection, the LCM of the two numbers is 20 and HCF is 4. And the product 80 of the two numbers equals the product of LCM and HCF which is 4x20 = 80 .
Proof 2:
Let a and be be the two numbers whose values are to be found. Let a > b. The following properties of LCM and HCF are known:
LCM is greater than or equal to HCF.
LCM is always >= a, the biggest number in the present problem
HCF is always <= b, the smallest number in our case
LCM = 20, HCF = 4
Therefore, 20 >= a and 4 <= b
It follows from above that the
Two numbers are 20 and 4 (Proved)
Let us call the two numbers as first and second number and break down the L.C.M and H.C.F into their constituent prime factors
L.C.M=20=2×2×5
H.C.F=4=2×2
From the very definition, H.C.F. is the greatest number that divides both the undetermined numbers. Therefore, 2x2 are factors of the two numbers. This gives us 4 as one of the two unknown numbers.
From definition of L.C.M., the two unknown numbers divide the L.C.M. This means 2x2x5 (= 20) are factors of the two numbers. Of the 3 factors 2,2, and 5, two (2,2) have already been counted. We are left only with 5. Therefore, the second number is 4x5 = 20
Thus, the two numbers are: 20 and 4 (Proved)
Correctness of the result is easily verifiable since, by inspection, the LCM of the two numbers is 20 and HCF is 4. And the product 80 of the two numbers equals the product of LCM and HCF which is 4x20 = 80 .