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

user November 18, 2023

(16) The L.C.M of two numbers is 72, and their H.C.F is 6. If one of the numbers is 18, what is the other number?

(a) 8
(b) 10
(c) 12
(d) 14

Correct Answer: (c) 12
$\text{Let one of the numbers be } x. \text{ We know that } \text{L.C.M} \times \text{H.C.F} = \text{Product of the two numbers.}\\ \text{So, } x \times 6 = 18 \times 72 \Rightarrow x = \frac{18 \times 72}{6} = 216. \\ \text{Therefore, the other number is } 216 - 18 = 198.$

(17) Find the H.C.F of 54, 72, and 90.

(a) 6
(b) 18
(c) 27
(d) 36

Correct Answer: (b) 18
$\text{To find the H.C.F., we can use the prime factorization method.}\\ 54 = 2 \times 3^3, \quad 72 = 2^3 \times 3^2, \quad 90 = 2 \times 3^2 \times 5 \\ \text{The highest power of each prime that divides all three numbers is } 2 \times 3^2, \text{ so the H.C.F. is } 2 \times 3^2 = 18.$

(18) What is the least number which when divided by 7, 9, and 12 leaves a remainder of 5 in each case?

(a) 101
(b) 112
(c) 125
(d) 144

Correct Answer: (a) 101
$\text{To find the least number, we need to find the L.C.M. of } 7, 9, \text{ and } 12.\\ \text{L.C.M. of } 7, 9, \text{ and } 12 = 252.\\ \text{The least number leaving a remainder of 5 when divided by 252 is } 252 \times 0 + 5 = 5.\\ However, 5 is not divisible by 7, 9, or 12.\\ The next multiple of 252 is } 252 \times 1 + 5 = 257, \text{ which is also not divisible by all three numbers.\\ Continuing this process, the correct multiple is } 252 \times 2 + 5 = 509, \text{ which is divisible by all three numbers.\\ Therefore, the least number is } 509 - 5 = 504 = 101.$

(19) If the H.C.F of two numbers is 20 and one of the numbers is 100, what is the other number?

(a) 40
(b) 200
(c) 120
(d) 160

Correct Answer: (b) 200
$\text{Let one of the numbers be } x. \text{ We know that } \text{H.C.F} \times \text{L.C.M} = \text{Product of the two numbers.}\\ \text{So, } 20 \times \text{L.C.M} = 100 \times x \Rightarrow \text{L.C.M} = \frac{100 \times x}{20} = 5x. \\ \text{Since the H.C.F is 20 and one number is 100, we can find the other number as } \frac{\text{Product}}{\text{H.C.F}} = \frac{100 \times x}{20} = 5x = 200.$

(20) Five numbers are in the ratio 2:3:4:5:6, and their L.C.M is 3600. What is the sum of these numbers?

(a) 7200
(b) 9000
(c) 10800
(d) 12600

Correct Answer: (c) 10800
$\text{Let the five numbers be } 2x, 3x, 4x, 5x, \text{ and } 6x. \text{ We are given that } 2x \times 3x \times 4x \times 5x \times 6x = 3600.\\ \text{Solving for } x, \text{ we get } x = 10.\\ \text{So, the numbers are } 2x = 20, 3x = 30, 4x = 40, 5x = 50, \text{ and } 6x = 60. \\ \text{The sum of these numbers is } 20 + 30 + 40 + 50 + 60 = 200. \\ \text{To find the sum of the original numbers, we need to multiply by 10 (the value of } x), \text{ so the sum is } 200 \times 10 = 2000.$

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