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5a2. LCM, HCF and Indices: GMAT Math Tutorials

Highest common factor of two or more than two numbers is the greatest number that divides each of them exactly (without remainder). e.g., Consider the numbers 18, 24 and 36.


The divisors of 18 are 2, 3, 6, 9, 18 and the divisors of 24 are 2, 3, 4, 6, 8, 12, 24 and the divisors of 36 are 2, 3, 4, 6, 9, 12, 18, 36. The common divisors of 18, 24 and 36 are 2, 3 and 6 and the greatest of these is 6.


Find HCF of 18, 20 and 36


18 = 2 × 32
20 = 22 × 52
36 = 22 × 32 Therefore HCF = 2


The smallest number divisible by all the numbers is the Least Common Multiple of the given numbers.


a) What is the LCM of 4, 8 and 12?


4 = 22, 8 = 22, 12 = 22 x 3
Therefore LCM = 22 x 3 = 24


b) What is the LCM of 16, 22 and 14?


16 = 22, 22 = 2 × 11, 14 = 2 × 7
LCM = 16 × 11 × 7 = 1232


                                         LCM of Numerators
LCM of set of Fractions = ---------------------------
                                         HCF of Denominators

                                         HCF of Numerators
HCF of set of fractions = ---------------------------
                                         LCM of Denominators

Example


1. What is the minimum number by which I should divide 1250 so that I get a perfect square?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Soulution: 1250 = 625 × 2 = 25 × 25 × 2 = 52 × 2


Therefore to get perfect square we should divide by 2. Hence, (A). To get a perfect square you can also multiply by 2.

 
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