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

user November 18, 2023

(11) The L.C.M of two numbers is 60, and their H.C.F is 6. If one of the numbers is 18, what is the other number?

(a) 10
(b) 12
(c) 14
(d) 20

Correct Answer: (d) 20
$\text{Let one of the numbers be } x. \text{ We know that } \text{L.C.M} \times \text{H.C.F} = \text{Product of the two numbers.}\\ \text{So, } x \times 6 = 18 \times 60 \Rightarrow x = \frac{18 \times 60}{6} = 180. \\ \text{Therefore, the other number is } 180 - 18 = 162.$

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

(a) 6
(b) 8
(c) 10
(d) 12

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

(13) What is the least number which when divided by 5, 8, and 12 leaves a remainder of 3 in each case?

(a) 99
(b) 112
(c) 125
(d) 144

Correct Answer: (a) 99
$\text{To find the least number, we need to find the L.C.M. of } 5, 8, \text{ and } 12.\\ \text{L.C.M. of } 5, 8, \text{ and } 12 = 120.\\ \text{The least number leaving a remainder of 3 when divided by 120 is } 120 \times 1 + 3 = 123.\\ However, 123 is not divisible by 5, 8, or 12.\\ The next multiple of 120 is } 120 \times 2 + 3 = 243, \text{ which is also not divisible by all three numbers.\\ Continuing this process, the correct multiple is } 120 \times 3 + 3 = 363, \text{ which is divisible by all three numbers.\\ Therefore, the least number is } 363 - 3 = 360 = 99.$

(14) If the H.C.F of two numbers is 15 and one of the numbers is 75, what is the other number?

(a) 45
(b) 225
(c) 135
(d) 195

Correct Answer: (b) 225
$\text{Let one of the numbers be } x. \text{ We know that } \text{H.C.F} \times \text{L.C.M} = \text{Product of the two numbers.}\\ \text{So, } 15 \times \text{L.C.M} = 75 \times x \Rightarrow \text{L.C.M} = \frac{75 \times x}{15} = 5x. \\ \text{Since the H.C.F is 15 and one number is 75, we can find the other number as } \frac{\text{Product}}{\text{H.C.F}} = \frac{75 \times x}{15} = 5x = 225.$

(15) Four numbers are in the ratio 2:3:4:5, and their L.C.M is 1200. What is the sum of these numbers?

(a) 1440
(b) 1600
(c) 1800
(d) 2000

Correct Answer: (c) 1800
$\text{Let the four numbers be } 2x, 3x, 4x, \text{ and } 5x. \text{ We are given that } 2x \times 3x \times 4x \times 5x = 1200.\\ \text{Solving for } x, \text{ we get } x = 5.\\ \text{So, the numbers are } 2x = 10, 3x = 15, 4x = 20, \text{ and } 5x = 25. \\ \text{The sum of these numbers is } 10 + 15 + 20 + 25 = 70.$

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