Page 25 - classs 6 a_Neat

CONCEPT OF POWER EXAMPLE 6: Find the HCF of 144 and 198 by the prime faccorizalion method.
SOLUTION: We have:
We write, 2 × 2 = 22 (read as 2 raised to the power 2), 2 144 2 198
2 × 2 × 2 = 23 (read as 2 raised to the power 3), 2 72 3 99
2 × 2 × 2 × 2 = 24 (read as 2 raised to the power 4).
and so on. 2 36 3 33
2 18 11 11
Similarly, 3 × 3 = 3 2, 3 × 3 × :3 = 3 ‘. 3 × 3 × 3 × 3 = 3 4• and 3 9 1
so on. In general. a × a × ... taken m times = am . 3 3
1
EXAMPLE 3: Give the prime factorization of 1260. 144 = 2 × 2 × 2 × 2 × 3 × 3 = 2 × 3 .
2
4
2 1260 And, 198 = 2 × 3 × 3 × 11 = 2 × 3 × 11.
2
2 630 HCF of 144 and 198 = 2 × 3 = 18.
2
3 315
3 105 EXAMPLE 7: Find the HCF of 396 and 1080 by the prime factorization method.
SOLUTION: Method
5 35
7 7 2 396 2 1080
1 2 196 2 540
1260 = 2 × 2 × 3 × 3 × 7 = 2 × 3 × 5 × 7. 3 99 2 270
2
2
EXAMPLE 4: Give the prime factorization of 20570. 3 33 5 135
SOLUTION: We have:
2 20570 11 11 3 27
5 10285 1 3 9
11 2057 3 3
11 187 1
17 17 So, 396 = 2 × 3 × 11.

2
2
1 And, 1080 = 2 × 3 × 5.
3
3
Therefore, 20570 = 2 × 5 × 11 × 11 × 17 = 2 × 5 × 112 × 17. Hence the HCF of 396 and 1080 is 2 × 3 = 36.
2
2
EXAMPLE 8: Find the HCF of 144,180 and 192 by the prime factorization method.
HCF AND LCM
2 144 2 180 2 192
HIGHEST COMMON FACTOR (HCF) The greatest number which is a common factor of two or more 2 72 2 90 2 96
given numbers, is called their highest commonfactor or greatest common divisor or greatest common 2 36 3 45 2 48
measure, written as HCF or GCD or GCM. 2 18 3 15 2 24

3 9 5 5 2 12
EXAMPLE 5: Let us find the HCF of 24 and 32.
SOLUTION: All the factors of 24 are: [1,2,3,4,6,8,12,24] 3 3 1 2 6
All the factors of 32 are: [1, 2, 4, 8, 16, 32 ] 1 3 3
Common factors of 24 and 32 are: [ 1, 2, 4, 8 ] 4 2 1
Thus, the highest common factor of 24 and 32 is 8. So, 144 = 2 × 2 × 2 × 2 × 3 × 3 = 2 × 3 ;
2
2
Hence, HCF of 24 and 32 = 8. 180 = 2 × 2 × 3 × 3 × 5 = 2 × 3 × 5;
192 = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 2 × 3.
6
Therefore, the HCF of the given numbers = 2 × 3 = 12
2
TO FIND HCF (BY PRIME FACTORIZATION METHOD)
TO FIND HCF (BY DIVISION METHOD)
We first find the prime factorization of each of the given numbers. Then, the product of all common primef ac- Suppose two numbers are given. Divide the greater number by the smaller one. Next, divide the divisor by the
tors, using the least power of each common prime factor, is the HCF of the given numbers. remainder. Go on repeating the process of dividing the preceding divisor by the remainder last obtained till the
remainder zero ts obtained. Then the last divisor ts the required HCF of the given numbers.