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Question:

Find the LCM and HCF of the following pairs of integers and verify that LCM ร HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Answer:

**Solution:**

**(i) 26 and 91 **

**The prime factors of 26 and 91 are : **

**26 = 2 x 13**

**91= 7 x 13**

**We know that LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with the highest power.**

** L.C.M (26,91) = 2 x 7 x 13**

**Hence, L.C.M of (26, 91) = 182**

**H.C.F of (26, 91) = 13**

**Since, Highest Common Factor of two or more numbers = Product of the smallest Power of each common prime factor involved in the numbers.**

**We know that L.C.M x H.C.F = First number x Second number**

**182 x 13 = 26 x 91**

**2366 = 2366. Verified**

**(ii) 510 and 92**

**The prime factors of 510 and 92 are :**

**510 = 2 x 3 x 5 x 17**

**92 = 2 x 2 x 23 = 2² × 23**

**We know that LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with the highest power.**

** L.C.M ( 510 ,92) = 2 x 2 x 3 x 5 x 23 x 17**

**L.C.M (510 ,92) = 23460**

**H.C.F (510 , 92) = 2**

**Since, HCF of two or more numbers= Product of the smallest Power of each common prime factor involved in the numbers.**

**We know that L.C.M x H.C.F = First Number x Second Number**

**23460 x 2 = 510 x 92**

**46920 = 46920. verified.**

**(iii) 336 and 54**

**The prime factors of 336 and 54 are :**

**336 = 2 x 2 x 2 x 2 x 3 x 7 = 2โด × 3¹ × 7¹**

**54 = 2 x 3 x 3 x 3 = 2¹ × 3³**

**We know that LCM of two or more numbers = product of the greatest power of each prime factor involved in the numbers, with the highest power.**

** L.C.M (336 , 54) = 2โด × 3³× 7 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 7**

**Hence, L.C.M( 336 , 54) = 3024**

**H.C.F (336 , 54) = 2 × 3**

**H.C.F (336 , 54) = 6**

**Since, HCF of two or more numbers = Product of the smallest Power of each common prime factor involved in the numbers.**

**We know that L.C.M x H.C.F = First Number x Second Number**

**3024 x 6 = 336 x 54**

**18144 = 18144. Verified**

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