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3. Find the HCF of the following numbers using the prime factorisation method. EXAMPLE 3: Find the LCM of 112, 168, 266 by the prime factorization method.
a. 27,81 b. 18, 117 c. 250,450 SOLUTION: We Have
d. 256, 1,024 e. 180,270,630 2 112 2 168 2 266
2 56 2 84 7 133
4. Find the HCF of the following using the long division method. 2 28 2 42 19 19
a. 144,372 b. 575,920 c. 891, 1377 2 14 3 21 1
d. 830, 790,650 e. 2,241, 8,217, 747 7 7 7 7 112 = 2 × 7
4
3
1 1 168 = 2 × 3 × 7
5.Find the greatest number that divides 43 and 61 leaving a remainder of 1 in each case. 266 = 2 × 7 × 19
6.Find the largest number that divides 246 and 1,030 leaving a remainder of 6 in each case.
7.Find the largest number that divides 1,38S and 1,4S7 leaving remainders of Sand 2, respectively. Therefore. the LCM of the given numbers = 2 × 3 × 7 × 19 = 6384.
4
8. Find the greatest number that divides 131, 160 and 223 leaving remainders of 7, Sand 6, respectively.
9. Two tankers contain 960 litres and 1,024 litres of petrol, respectively. Find the capacity of the TO FIND LCM (BY DIVISION METHOD) In this method, we arrange the given numbers in a line, in
container that can measure the petrol in either tanker an exact number of times. any order. We divide by a number which divides exactly at least two of the given numbers and
10. The length of two rods is 7 m so cm and 10 m SO cm, respectively. Find the length of the longest carryforward the numbers which are not divisible. This process is repeated till no two of the given
tape that can measure these lengths exactly. numbers are disible by a common number. The product of the divisors and the undivided numbers
is rhe required LCM if the given numbers.
LOWEST COMMON MULTIPLE (LCM) The lowest common multiple of two or more numbers is the
smallest number which is a multiple of each of the numbers. EXAMPLE 4: Find the LCM of 12. 15, 20. 27 by the division method.
SOLUTION: We have
EXAMPLE 1: Let us find the LCM of 4 and 6. 3 12, 15, 20, 27
SOLUTION: Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36,.... 4 4, 5, 20, 9
5 1, 5, 5, 9
Multiples of 6 are: 6. 12. 18. 24. 30. 36, .. .
1, 1, 1, 9
Common multiples of 4 and 6 are: 12. 24, 36, ... Hence. the LCM of the given numbers= 3 × 4 × 5 × 9 = 540.
Lowest common multiple of 4 and 6 is 12.
Hence. LCM of 4 and 6 = 12. EXAMPLE 5: Find the LCM of 22, 54, 108, 135 and 198.
SOLUTION: We have
TO FIND LCM (BY PRIME FACTORIZATION METHOD) In order tofind the LCM of two or 2 22, 54, 108, 135, 198
more given numbers we write the prime factorization of each of the given numbers. Then, 11 11.27, 54,135.99
the required LCM of these numbers is the product of all dijferent prime factors of 9 1. 27, 54, 135.
9 3 1, 3, 6. 15, 1
EXAMPLE 2: Find the LCM of 24, 36 and 40 by the prime factorization method. 1. 1, 2, 5, 1
SOLUTION: We Have
2 24 2 36 2 40 Hence, the LCM of the given numbers = 2 × 11 × 9 × 3 × 2 × 5 = 5940.
2 12 2 18 2 20
2 6 3 9 2 10 EXAMPLE 6: Find the smallest number which when diminished by 3 ts divisible by 21, 28, 36 and 45.
SOLUTION: We know that the smallest number divisible by 21, 28, 36 and 45 is their LCM. We
3 3 3 3 5 5 calculate this LCM as under:

1 1 1 7 21, 28, 36, 45
24 = 2 × 3 3 3, 4, 36, 45
3
36 = 2 × 3 2 3 1, 4, 12, 15
2
40 = 2 × 5 4 1, 4, 4, 5
3
Hence, the LCM of 24, 36 and 40 is 2 × 3 × 5 = 360. 1,1,1,5
3
2
Hence. the LCM of 21, 28, 36 and 45 is 7 × 3 × 3 × 4 × 5 = 1260.
Hence, the required number= (1260 + 3) = 1263.