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Answer:

The relation between the LCM and HCF of 106, 159 and 265 is given as, HCF(106, 159, 265) = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(106, 265)]

⇒ Prime factorization of 106, 159 and 265:

106 = 2 × 53

159 = 3 × 53

265 = 5 × 53

∴ LCM of (106, 159), (159, 265), (106, 265), and (106, 159, 265) is 318, 795, 530, and 1590 respectively.

Now, LHS = HCF(106, 159, 265) = 53.

And, RHS = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(106, 265)] = [(4466310) × 1590]/[318 × 795 × 530]

LHS = RHS = 53.

Hence verified.

Example 2: Find the highest number that divides 106, 159, and 265 completely.

Solution:

The highest number that divides 106, 159, and 265 exactly is their highest common factor.

Factors of 106 = 1, 2, 53, 106

Factors of 159 = 1, 3, 53, 159

Factors of 265 = 1, 5, 53, 265

The HCF of 106, 159, and 265 is 53.

∴ The highest number that divides 106, 159, and 265 is 53.

Example 3: Calculate the HCF of 106, 159, and 265 using LCM of the given numbers.

Solution:

Prime factorization of 106, 159 and 265 is given as,

106 = 2 × 53

159 = 3 × 53

265 = 5 × 53

LCM(106, 159) = 318, LCM(159, 265) = 795, LCM(265, 106) = 530, LCM(106, 159, 265) = 1590

⇒ HCF(106, 159, 265) = [(106 × 159 × 265) × LCM(106, 159, 265)]/[LCM(106, 159) × LCM (159, 265) × LCM(265, 106)]

⇒ HCF(106, 159, 265) = (4466310 × 1590)/(318 × 795 × 530)

⇒ HCF(106, 159, 265) = 53.

Therefore, the HCF of 106, 159 and 265 is 53.

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FAQs on HCF of 106, 159 and 265

What is the HCF of 106, 159 and 265?

Which of the following is HCF of 106, 159 and 265? 53, 273, 304, 270, 275

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What is the Relation Between LCM and HCF of 106, 159 and 265?

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