Question

(a) Write 216 as a product of its prime factors. Two positive integers are each greater than 25. Their lowest common multiple (LCM) is 216. Their highest common factor (HCF) is 18. Find the two integers.​

Answer 1

Answer:

Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216

Prime Factorization of 216: 216 = 2 × 2 × 2 × 3 × 3 × 3

Answer 2

The two numbers are 54 and 72.

Given:

Two positive integers  say a and b, are each greater than 25.

Their lowest common multiple (LCM) is 216.

Their highest common factor (HCF) is 18.

To Find:

I) Prime factorization of 216.

II) a and b.

Solution:

I) Prime factorization of 216 is

216 = 2×2×2×3×3×3= 216  = $2^{3}$× $3^{3}$

II) We have been given that LCM (a, b)= 216 and HCF (a, b)= 18

LCM of two numbers consists of unique factors of both the  numbers and HCF comprises of common factors of both the numbers.

Hence HCF (a, b)= 18= (2×3×3) is a factor of both a and b.

Now factorizing the LCM

LCM (a, b)= 216 = 2×2×2×3×3×3

LCM (a, b)= 2×2×(2×3×3)×3 = 2×2×18×3

LCM (a, b)= 2×2×3×18

Hence the possibilities of a and b are:

18×2=36

18×3= 54

18×(2×2)= 72

18×(2×3)= 108

18×(2×2×3)=216

But we know that,

LCM (a, b)× HCF (a, b)= a × b

Hence the only possibilities of of two such numbers satisfying the above condition are 54 and 72.

Hence the two numbers are 54 and 72.

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