Question 1 : Express each number as a product of its prime factors:
(1) 140
(2) 156
(3) 3825
(4) 5005
(5) 7429
Solutions :
(1) 140
To get the product of prime factor we will take it’s LCM. Therefore
140 =2*2*5*7*1 = 2^2*5*7.
(2) 156
By taking the LCM, we will get the product of its prime factor.
Hence, 156 = 2*2*13*3*1 = 2^2*13*3.
(3) 3825
To get the product of it’s prime factor , we will take the LCM.
Hence, 3825 =3*3*5*5*71*1 = 3^ 2*5^ 2*17.
(4) 5005
We will take the LCM of 5005 and we will get the product of prime factor.
Hence, 5005 = 5*7*11*13*1 = 5*7*11613.
(5) 7429
Now by taking the LCM of 7429, we will get the product of its prime factor.
Hence, 7429 = 17*19*23*1 =17*19*23.
Question 2 :
Find the LCM and HCF of the following pairs of integers and verify that LCM*HCF = product of the two numbers.
(1) 26 and 91
(2) 510 and 92
(3) 336 and 54
Solutions :
(1) 26 and 91
First we express 26 and 91 as product of its prime factor, we get
26 =2*13*1
91 = 7*13*1
Therefore, LCM (26,91) = 2*7*13*1 = 182
And HCF (26,91) = 13.
Verification
Now product of 26 and 91 = 2366
And product of LCM and HCF = 182*13 = 2366
Here we can see that
LCM*HCF = product if 26 and 91.
(2) 510 and 92
Again expressing the numbers in terms of their prime factor. We will get,
510 = 2*3*17*5*1
92 = 2*2*23*1
Therefore LCM = 2*2*3*5*17*23 = 23460
And HCF = 2
Verification :
Product of numbers (510,92) = 46920
Product of LCM and HCF = 23460*2 = 46920.
Hence it is proved that,
LCM*HCF = product of 510 and 92.
(3) 336 and 54
Expressing the numbers as product of their prime factor.we get,
336 = 2*2*2*2*7*3*1
54 = 2*3*3*3*1
Now LCM (336,54) = 2^4*3^3*7 = 3024
And HCF (336,54) = 2*3 = 6.
Verification :
Product of numbers = 18144
Product of LCM and HCF = 18144
Hence, LCM *HCF = product of 336 and 54.
