Question

Find theHCF of 106,159,265

Answer 1

ANSWER:

HCF of 106, 159 and 265 is the largest possible number that divides 106, 159 and 265 exactly without any remainder. The factors of 106, 159 and 265 are (1, 2, 53, 106), (1, 3, 53, 159) and (1, 5, 53, 265) respectively. There are 3 commonly used methods to find the HCF of 106, 159 and 265 - prime factorization, long division, and Euclidean algorithm.

Step-by-step explanation:

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.

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HCF of 106, 159 and 265

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HCF of 106, 159 and 265 is the largest possible number that divides 106, 159 and 265 exactly without any remainder. The factors of 106, 159 and 265 are (1, 2, 53, 106), (1, 3, 53, 159) and (1, 5, 53, 265) respectively. There are 3 commonly used methods to find the HCF of 106, 159 and 265 - prime factorization, long division, and Euclidean algorithm.

What is HCF of 106, 159 and 265?

Answer: HCF of 106, 159 and 265 is 53.

HCF of 106, 159 and 265

Explanation:

The HCF of three non-zero integers, x(106), y(159) and z(265), is the highest positive integer m(53) that divides x(106), y(159) and z(265) without any remainder.

Methods to Find HCF of 106, 159 and 265

Let's look at the different methods for finding the HCF of 106, 159 and 265.

Long Division Method

Prime Factorization Method

Listing Common Factors

HCF of 106, 159 and 265 by Long Division

HCF of 106, 159 and 265 can be represented as HCF of (HCF of 106, 159) and 265. HCF(106, 159, 265) can be thus calculated by first finding HCF(106, 159) using long division and thereafter using this result with 265 to perform long division again.

Step 1: Divide 159 (larger number) by 106 (smaller number).

Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (106) by the remainder (53). Repeat this process until the remainder = 0.

⇒ HCF(106, 159) = 53.

Step 3: Now to find the HCF of 53 and 265, we will perform a long division on 265 and 53.

Step 4: For remainder = 0, divisor = 53 ⇒ HCF(53, 265) = 53

Thus, HCF(106, 159, 265) = HCF(HCF(106, 159), 265) = 53.

HCF of 106, 159 and 265 by Prime Factorization

Prime factorization of 106, 159 and 265 is (2 × 53), (3 × 53) and (5 × 53) respectively. As visible, 106, 159 and 265 have only one common prime factor i.e. 53. Hence, the HCF of 106, 159 and 265 is 53.

HCF of 106, 159 and 265 by Listing Common Factors

HCF of 106, 159 and 265 by Listing Common Factors

Factors of 106: 1, 2, 53, 106

Factors of 159: 1, 3, 53, 159

Factors of 265: 1, 5, 53, 265

There are 2 common factors of 106, 159 and 265, that are 1 and 53. Therefore, the highest common factor of 106, 159 and 265 is 53.

☛ Also Check:

HCF of 34 and 85 = 17

HCF of 6, 72 and 120 = 6

HCF of 32 and 56 = 8

HCF of 54, 288 and 360 = 18

HCF of 2923 and 3239 = 79

HCF of 180, 252 and 324 = 36

HCF of 4 and 8 = 4

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HCF of 106, 159 and 265 Examples

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.

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.

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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