Quantitative Aptitude
HCF and LCM

For coprime numbers, HCF = 1. So LCM = Product. 143 = 11 × x. Therefore x = 13. Check: HCF(11,13) = 1 ✓

When two numbers share a common HCF (Highest Common Factor), both numbers must be multiples of that HCF. We can use this property along with the given sum to find the unknown number.

Step 1: Express both numbers as multiples of HCF

Since HCF = 11, both numbers can be written as:

where \(a\) and \(b\) are coprime integers (HCF of \(a\) and \(b\) is 1).

Step 2: Find the value of \(a\) using the known number

One number is 33, so:

Step 3: Use the sum condition to find \(b\)

The sum of both numbers is 99:

Step 4: Calculate the other number

Verification: HCF(33, 66) = 33... Wait, let me recalculate: 33 = 3 × 11 and 66 = 6 × 11 = 2 × 3 × 11. HCF = 11 ✓ and 33 + 66 = 99 ✓

Answer: The other number is 66 (Option B)

MP = 500 × 1.60 = ₹800. SP = 800 × 0.75 = ₹600. Profit = 100. Profit% = (100/500) × 100 = 20%

Let numbers be 2k, 3k, 4k. LCM(2k, 3k, 4k) = 12k = 240, so k = 20. HCF = k = 20

Using HCF × LCM = Product of two numbers. 12 × 180 = 36 × x. Therefore x = 2160/36 = 60

To find when all three bells ring together, we need the Least Common Multiple (LCM) of their ringing intervals.

Step 1: Find prime factorization of each interval

Step 2: Identify highest powers of each prime factor

For LCM, take the highest power of each prime that appears:

- Highest power of 2: \(2^3\) (from 8)

- Highest power of 3: \(3^2\) (from 18)

Step 3: Calculate the LCM

Step 4: Verify the answer

- \(72 \div 8 = 9\) ✓ (Bell 1 rings 9 times)

- \(72 \div 12 = 6\) ✓ (Bell 2 rings 6 times)

- \(72 \div 18 = 4\) ✓ (Bell 3 rings 4 times)

All three bells divide evenly into 72 minutes, confirming they ring together at this time.

Answer: The bells will ring together after \(72\) minutes (Option D)

Profit = 20% of CP; ₹200 = 0.2×CP; CP = ₹1000; SP = 1000 + 200 = ₹1200

After 15% discount: 85%; After 10% on that: 85×90/100 = 76.5%; Single discount = 100-76.5 = 23.5%

Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36; Time = 36/5 = 7.2 days

To find the LCM of 48, 64, and 96, we express each number as a product of prime factors, then take the highest power of each prime that appears.

Step 1: Prime factorization of each number

Divide each number by its prime factors:

Step 2: Identify all prime factors

The prime factors present are: \(2\) and \(3\)

Step 3: Take the highest power of each prime

- Highest power of \(2\): \(2^6\) (from 64)

- Highest power of \(3\): \(3^1\) (from 48 and 96)

Step 4: Calculate the LCM

Answer: The LCM of 48, 64, and 96 is \(192\) (Option A)