#### Mathematics (NTSE/Olympiad)

# Real Numbers

__The Fundamental Theorem of Arithmetic to Find H.C.F. and L.C.M.__

**Example : Find the L.C.M. and H.C.F. of the following pairs of integers by applying the Fundamental theorem of Arithmetic Method (Using the Prime Factorisation Method).**

(i) 26 and 91

(ii) 1296 and 2520

(iii) 17 and 25

Solution:

(i) Since, 26 = 2 × 13 and, 91 = 7 × 13

∴ L.C.M. = Product of each **prime factor** with highest powers = 2 × 13 × 7 = 182. (Answer)

i.e., L.C.M. (26, 91) = 182. (Answer)

H.C.F. = Product of common prime factors with lowest powers. = 13.

i.e., H.C.F (26, 91) = 13.

(ii) Since, 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2^{4} × 3^{4} and, 2520 = 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2^{3} × 3^{2} × 5 × 7

∴L.C.M. = Product of each prime factor with highest powers = 2^{4} × 3^{4} × 5 × 7 = 45,360

i.e., L.C.M. (1296, 2520) = 45,360 (Answer)

H.C.F. = Product of common prime factors with lowest powers. = 2^{3} × 3^{2} = 8 × 9 = 72

i.e., H.C.F. (1296, 2520) = 72. (Answer)

(iii) Since, 17 = 17 and, 25 = 5 × 5 = 5^{2}

∴ L.C.M. = 17 × 5^{2} = 17 × 25 = 425 and, H.C.F. = Product of common prime factors with lowest powers = 1, as given numbers do not have any common prime factor.

**In example 19 (i) : **Product of given two numbers = 26 × 91 = 2366 and, product of their L.C.M. and H.C.F. = 182 × 13 = 2366

∴ Product of L.C.M and H.C.F of two given numbers = Product of the given numbers

**In example 19 (ii) : **Product of given two numbers = 1296 × 2520 = 3265920 and, product of their L.C.M. and H.C.F. = 45360 × 72 = 3265920

∴L.C.M. (1296, 2520) × H.C.F. (1296, 2520) = 1296 × 2520

**In example 19 (iii) :**

The given numbers 17 and 25 do not have any common prime factor. Such numbers are called co-prime numbers and their H.C.F. is always equal to 1 (one), whereas their L.C.M. is equal to the product of the numbers. But in case of two co-prime numbers also, the product of the numbers is always equal to the product of their L.C.M. and their H.C.F.

As, in case of co-prime numbers 17 and 25;

H.C.F. = 1; L.C.M. = 17 × 25 = 425;

Product of numbers = 17 × 25 = 425 and product of their H.C.F. and L.C.M. = 1 × 425 = 425.

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