*Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.*

*The prime factorisation of a natural number is unique, except for the order of its factors.*

⊕ Property of HCF and LCM of two positive integers ‘a’ and ‘b’:*HCF(a,b) x LCM(a,b) = a x b*

**PRIME FACTORISATION METHOD TO FIND HCF AND LCM**

HCF(a, b) = Product of the smallest power of each common prime factor in the numbers.

LCM(a, b) = Product of the greatest power of each prime factor, involved in the numbers.

**Q.1.** Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

**Q.2. **Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

**Q.3. **Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product

of the two numbers: (i) 26 and 91 (ii) 336 and 54

**Q.4.** Find the LCM and HCF of the following integers by applying the prime factorisation method: (i)

12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

**Q.5.** Explain why 3 × 5 × 7 + 7 is a composite number.

**Q.6. **Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

**Q.7. **Can the number 4n, n being a natural number, end with the digit 0? Give reasons.

**Q.8. **Given that HCF (306, 657) = 9, find LCM (306, 657).

**Q.9. **If two positive integers a and b are written as a = x^{3}y^{2} and b = xy^{3}; x, y are prime numbers, then

find the HCF (a, b).

**Q.10. **If two positive integers p and q can be expressed as p = ab^{2} and q = a^{3}b; a, b being prime

numbers, then find the LCM (p, q).

**Q.11. **Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

**Q.12.** Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7

respectively.

**Q.13.** Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum

capacity of a container which can measure the kerosene oil of both the tankers when used an

exact number of times.

**Q.14.** In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm

respectively. What is the minimum distance each should walk so that all can cover the same

distance in complete steps?

**Q.15. **Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each

case.

**Q.16. **The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the

longest tape which can measure the three dimensions of the room exactly.