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

(31) The L.C.M of two numbers is 72, and their H.C.F is 8. If one of the numbers is 24, what is the other number?

• (a) 8
• (b) 12
• (c) 216
• (d) 504

Answer

Correct Answer: (c) 216
\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.}\\<br /> \text{So, } x \times 8 = 24 \times 72 \Rightarrow x = \frac{24 \times 72}{8} = 216. \\<br /> \text{Therefore, the other number is } 216.

(32) Find the H.C.F of 54, 90, and 126.

• (a) 6
• (b) 9
• (c) 12
• (d) 18

Answer

Correct Answer: (d) 18
\text{To find the H.C.F., we can use the prime factorization method.}\\<br /> 54 = 2 \times 3^3, \quad 90 = 2 \times 3^2 \times 5, \quad 126 = 2 \times 3^2 \times 7 \\<br /> \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.

(33) What is the least number which when divided by 4, 6, and 8 leaves a remainder of 2 in each case?

• (a) 20
• (b) 50
• (c) 90
• (d) 120

Answer

Correct Answer: (b) 50
\text{To find the least number, we need to find the L.C.M. of } 4, 6, \text{ and } 8.\\<br /> \text{L.C.M. of } 4, 6, \text{ and } 8 = 24.\\<br /> \text{The least number leaving a remainder of 2 when divided by 24 is } 24 \times 0 + 2 = 2.\\<br /> However, 2 is not divisible by 4, 6, or 8.\\<br /> The next multiple of 24 is } 24 \times 1 + 2 = 26, \text{ which is also not divisible by all three numbers.\\<br /> Continuing this process, the correct multiple is } 24 \times 2 + 2 = 50, \text{ which is divisible by all three numbers.\\<br /> Therefore, the least number is } 50.

(34) If the H.C.F of two numbers is 36 and one of the numbers is 216, what is the other number?

• (a) 54
• (b) 72
• (c) 108
• (d) 144

Answer

Correct Answer: (c) 108
\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.}\\<br /> \text{So, } 36 \times \text{L.C.M} = 216 \times x \Rightarrow \text{L.C.M} = \frac{216 \times x}{36} = 6x. \\<br /> \text{Since the H.C.F is 36 and one number is 216, we can find the other number as } \frac{\text{Product}}{\text{H.C.F}} = \frac{216 \times x}{36} = 6x = 108.

(35) Eight numbers are in the ratio 3:4:5:6:7:8:9:10, and their L.C.M is 5040. What is the sum of these numbers?

• (a) 4320
• (b) 4800
• (c) 5600
• (d) 6240

Answer

Correct Answer: (d) 6240
\text{Let the eight numbers be } 3x, 4x, 5x, 6x, 7x, 8x, 9x, \text{ and } 10x. \text{ We are given that } 3x \times 4x \times 5x \times 6x \times 7x \times 8x \times 9x \times 10x = 5040.\\<br /> \text{Solving for } x, \text{ we get } x = 5.\\<br /> \text{So, the numbers are } 3x = 15, 4x = 20, 5x = 25, 6x = 30, 7x = 35, 8x = 40, 9x = 45, \text{ and } 10x = 50. \\<br /> \text{The sum of these numbers is } 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = 260. \\<br /> \text{To find the sum of the original numbers, we need to multiply by 5 (the value of } x), \text{ so the sum is } 260 \times 5 = 1300.