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Which of the following statements are correct A. The product of two integers is 500 and their difference is 5. Then, one of the numbers is 25. B. The HCF of 108, 288 and 360 is 18. C. The least number which is exactly divisible by each one of the given numbers is called their LCM. Choose the correct answer from the options given below:?

LohaHridoy
  1. A and B only
  2. A and C only
  3. B and C only
  4. C only

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1 Answers

AhmedNazir

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Option 2 : A and C only

Calculation:

Statement A:

Traditional Method:

Let the two integers be a and b.

It is given that the product of two integers is 500 and their difference is 5, so equation will be

⇒ a × b = 500 ............(i)

⇒ a - b = 5 .................(ii)

We know that

(a + b)2 = (a - b)2 + 4ab

⇒ (a + b)2 = (5)2 + 4 × 500

⇒ (a + b)2 = 25 + 2000

⇒ (a + b)2 = 2025 = 452

⇒ (a + b) = 45 .............(iii)

On solving equation (i) and (iii) we get,

a = 25 and b = 20

Alternate MethodWe can also find the two integers by assuming such two numbers whose product is 500 and difference is 5.

⇒ 500 = 5 × 100        (100 - 5 ≠ 5)

⇒ 500 = 50 × 10        (50 - 10 ≠ 5)

⇒ 500 = 25 × 20        (25 - 20 = 5)

Hence, the numbers are 25 and 20

Statement A is correct.

Statement B:

By Prime Factorization method, the HCF of 108, 288 and 360 is

⇒ 108 = 2 × 2 × 3 × 3 × 3

⇒ 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3

⇒ 360 = 2 × 2 × 2 × 3 × 3 × 5

Here, common prime factors of 108, 288 and 360 = 2 × 2 × 3 × 3 = 36

Statement B is incorrect.

Statement C:

Least Common Multiple(LCM) = The least number which is exactly divisible by each one of the given numbers is called their LCM.

Let a and b are two integers, then their LCM is,

LCM(a, b) = (a × b)/HCF of a and b.

Statement C is correct.

∴ Only statement A and C are correct.

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