22. Which pair of numbers has a HCF that is not a prime number?
(1) 60, 231
(2) 15, 80
(3) 24, 26
(4) 30, 42
27. The maximum number of students among whom 1001 pens and 910 pencils can be distributed, such that each student gets the same number of pens and same number of pencils is:
(1) 910
(2) 1001
(3) 91
(4) 191
30. Which of the following is the LCM of 9, 14 and 21?
(1) 189
(2) 126
(3) 252
(4) 378
Answer (1)
Let's address each question one by one:
Which pair of numbers has a HCF that is not a prime number? To find the Highest Common Factor (HCF), we need to determine the greatest number that divides both numbers of the pair completely.
(1) 60 and 231:
Factors of 60: 2 2 × 3 × 5
Factors of 231: 3 × 7 × 11
HCF of 60 and 231 is 3 (a prime number).
(2) 15 and 80:
Factors of 15: 3 × 5
Factors of 80: 2 4 × 5
HCF of 15 and 80 is 5 (a prime number).
(3) 24 and 26:
Factors of 24: 2 3 × 3
Factors of 26: 2 × 13
HCF of 24 and 26 is 2 (a prime number).
(4) 30 and 42:
Factors of 30: 2 × 3 × 5
Factors of 42: 2 × 3 × 7
HCF of 30 and 42 is 6 (not a prime number).
The correct option is (4) 30 and 42.
The maximum number of students among whom 1001 pens and 910 pencils can be evenly distributed. This involves finding the HCF of 1001 and 910, as it will give the number of students that can be evenly distributed.
Prime factorization of 1001: 7 × 11 × 13
Prime factorization of 910: 2 × 5 × 7 × 13
Common factors: 7 × 13 = 91
The correct option is 91.
Which of the following is the LCM of 9, 14, and 21? To find the Least Common Multiple (LCM), use the highest powers of all prime factors present.
Prime factors of 9: 3 2
Prime factors of 14: 2 × 7
Prime factors of 21: 3 × 7
LCM: 2 1 × 3 2 × 7 1 = 126
The correct option is (2) 126.
