Problems on H.C.F and L.C.M

user November 18, 2023

(26) The L.C.M of two numbers is 36, and their H.C.F is 6. If one of the numbers is 18, what is the other number?

(a) 12
(b) 18
(c) 72
(d) 144

Correct Answer: (c) 72
$\text{Let one of the numbers be } x. \text{ We know that } \text{L.C.M} \times \text{H.C.F} = \text{Product of the two numbers.}\\ \text{So, } x \times 6 = 18 \times 36 \Rightarrow x = \frac{18 \times 36}{6} = 108. \\ \text{Therefore, the other number is } 108 - 18 = 90.$

(27) Find the H.C.F of 36, 48, and 60.

(a) 6
(b) 12
(c) 18
(d) 24

Correct Answer: (b) 12
$\text{To find the H.C.F., we can use the prime factorization method.}\\ 36 = 2^2 \times 3^2, \quad 48 = 2^4 \times 3, \quad 60 = 2^2 \times 3 \times 5 \\ \text{The highest power of each prime that divides all three numbers is } 2^2 \times 3, \text{ so the H.C.F. is } 2^2 \times 3 = 12.$

(28) What is the least number which when divided by 5, 7, and 9 leaves a remainder of 3 in each case?

(a) 18
(b) 33
(c) 45
(d) 93

Correct Answer: (d) 93
$\text{To find the least number, we need to find the L.C.M. of } 5, 7, \text{ and } 9.\\ \text{L.C.M. of } 5, 7, \text{ and } 9 = 315.\\ \text{The least number leaving a remainder of 3 when divided by 315 is } 315 \times 0 + 3 = 3.\\ However, 3 is not divisible by 5, 7, or 9.\\ The next multiple of 315 is } 315 \times 1 + 3 = 318, \text{ which is also not divisible by all three numbers.\\ Continuing this process, the correct multiple is } 315 \times 2 + 3 = 633, \text{ which is divisible by all three numbers.\\ Therefore, the least number is } 633 - 3 = 630 + 3 = 633.$

(29) If the H.C.F of two numbers is 30 and one of the numbers is 150, what is the other number?

(a) 50
(b) 60
(c) 70
(d) 90

Correct Answer: (d) 90
$\text{Let one of the numbers be } x. \text{ We know that } \text{H.C.F} \times \text{L.C.M} = \text{Product of the two numbers.}\\ \text{So, } 30 \times \text{L.C.M} = 150 \times x \Rightarrow \text{L.C.M} = \frac{150 \times x}{30} = 5x. \\ \text{Since the H.C.F is 30 and one number is 150, we can find the other number as } \frac{\text{Product}}{\text{H.C.F}} = \frac{150 \times x}{30} = 5x = 90.$

(30) Seven numbers are in the ratio 4:5:6:7:8:9:10, and their L.C.M is 2520. What is the sum of these numbers?

(a) 2520
(b) 2880
(c) 3150
(d) 3600

Correct Answer: (b) 2880
$\text{Let the seven numbers be } 4x, 5x, 6x, 7x, 8x, 9x, \text{ and } 10x. \text{ We are given that } 4x \times 5x \times 6x \times 7x \times 8x \times 9x \times 10x = 2520.\\ \text{Solving for } x, \text{ we get } x = 3.\\ \text{So, the numbers are } 4x = 12, 5x = 15, 6x = 18, 7x = 21, 8x = 24, 9x = 27, \text{ and } 10x = 30. \\ \text{The sum of these numbers is } 12 + 15 + 18 + 21 + 24 + 27 + 30 = 147. \\ \text{To find the sum of the original numbers, we need to multiply by 3 (the value of } x), \text{ so the sum is } 147 \times 3 = 441.$

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