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

(21) The L.C.M of two numbers is 48, and their H.C.F is 4. If one of the numbers is 12, what is the other number?

• (a) 6
• (b) 96
• (c) 120
• (d) 144

Answer

Correct Answer: (b) 96
\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 4 = 12 \times 48 \Rightarrow x = \frac{12 \times 48}{4} = 144. \\<br /> \text{Therefore, the other number is } 144 - 12 = 132.

(22) Find the H.C.F of 64, 80, and 96.

• (a) 4
• (b) 8
• (c) 12
• (d) 16

Answer

Correct Answer: (d) 16
\text{To find the H.C.F., we can use the prime factorization method.}\\<br /> 64 = 2^6, \quad 80 = 2^4 \times 5, \quad 96 = 2^5 \times 3 \\<br /> \text{The highest power of each prime that divides all three numbers is } 2^4, \text{ so the H.C.F. is } 2^4 = 16.

(23) What is the least number which when divided by 6, 8, and 10 leaves a remainder of 4 in each case?

• (a) 34
• (b) 40
• (c) 64
• (d) 84

Answer

Correct Answer: (c) 64
\text{To find the least number, we need to find the L.C.M. of } 6, 8, \text{ and } 10.\\<br /> \text{L.C.M. of } 6, 8, \text{ and } 10 = 120.\\<br /> \text{The least number leaving a remainder of 4 when divided by 120 is } 120 \times 0 + 4 = 4.\\<br /> However, 4 is not divisible by 6, 8, or 10.\\<br /> The next multiple of 120 is } 120 \times 1 + 4 = 124, \text{ which is also not divisible by all three numbers.\\<br /> Continuing this process, the correct multiple is } 120 \times 2 + 4 = 244, \text{ which is divisible by all three numbers.\\<br /> Therefore, the least number is } 244 - 4 = 240 + 4 = 244 = 64.

(24) If the H.C.F of two numbers is 24 and one of the numbers is 120, what is the other number?

• (a) 30
• (b) 45
• (c) 180
• (d) 240

Answer

Correct Answer: (c) 180
\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, } 24 \times \text{L.C.M} = 120 \times x \Rightarrow \text{L.C.M} = \frac{120 \times x}{24} = 5x. \\<br /> \text{Since the H.C.F is 24 and one number is 120, we can find the other number as } \frac{\text{Product}}{\text{H.C.F}} = \frac{120 \times x}{24} = 5x = 180.

(25) Six numbers are in the ratio 3:4:5:6:7:8, and their L.C.M is 3360. What is the sum of these numbers?

• (a) 2016
• (b) 2688
• (c) 3360
• (d) 4032

Answer

Correct Answer: (b) 2688
\text{Let the six numbers be } 3x, 4x, 5x, 6x, 7x, \text{ and } 8x. \text{ We are given that } 3x \times 4x \times 5x \times 6x \times 7x \times 8x = 3360.\\<br /> \text{Solving for } x, \text{ we get } x = 4.\\<br /> \text{So, the numbers are } 3x = 12, 4x = 16, 5x = 20, 6x = 24, 7x = 28, \text{ and } 8x = 32. \\<br /> \text{The sum of these numbers is } 12 + 16 + 20 + 24 + 28 + 32 = 132. \\<br /> \text{To find the sum of the original numbers, we need to multiply by 4 (the value of } x), \text{ so the sum is } 132 \times 4 = 528.