Need to find LCM of 8, 12, 16. 8 = 2³, 12 = 2² × 3, 16 = 2⁴. LCM = 2⁴ × 3 = 48 minutes.

So they ring again at 12:00 + 48 min = 12:48 PM.

Let numbers be 12a and 12b where HCF(a,b)=1. LCM = 12ab = 240, so ab = 20.

Numbers: 12a and 12b with |12a - 12b| = 12, so |a - b| = 1.

If a=4, b=5: numbers are 48 and 60.

Check: 48-60 = -12 (difference is 12).

For two numbers, the fundamental relationship is \(\text{HCF} \times \text{LCM} = \text{Product of the two numbers}\). We'll verify which pairs satisfy both the HCF = 7 and LCM = 140 conditions.

Step 1: Apply the HCF-LCM product rule

For any two numbers \(a\) and \(b\):

Given HCF = 7 and LCM = 140:

Step 2: Check option (A): 7 and 140

Option (A) is valid.

Step 3: Check option (B): 14 and 70

Option (B) is invalid.

Step 4: Check option (C): 28 and 35

Option (C) is valid.

Answer: Both options (A) and (C) satisfy the conditions HCF = 7 and LCM = 140 (Option D)

To find the LCM of three numbers, we use prime factorization and take the highest power of each prime factor.

Step 1: Prime factorization of each number

Step 2: Identify all prime factors and their highest powers

Across all three numbers:

- Highest power of 2: \(2^2\) (from 108)

- Highest power of 3: \(3^4\) (from 162)

- Highest power of 5: \(5^1\) (from 135)

Step 3: Calculate the LCM

Step 4: Final multiplication

Answer: The LCM of 108, 135, and 162 is \(1620\) (Option C)

Let numbers be 2k, 3k, 4k. HCF = k = 5.

Numbers are 10, 15, 20.

Sum = 10 + 15 + 20 = 45.

To find the smallest 4-digit number divisible by 12, 18, and 24, we need to find the least common multiple (LCM) and then scale it appropriately.

Step 1: Find the prime factorization of each number

Step 2: Calculate the LCM

For the LCM, take the highest power of each prime factor:

Step 3: Find the smallest 4-digit multiple of 72

The smallest 4-digit number is 1000. Divide it by 72:

Since we need a whole number of groups of 72, round up to the next integer: \(14\)

Step 4: Calculate the required number

Verify: \(1008 \div 12 = 84\), \(1008 \div 18 = 56\), \(1008 \div 24 = 42\) ✓

Answer: The smallest 4-digit number divisible by 12, 18, and 24 is \(1008\) (Option A)

When three numbers are in a given ratio, we can express them using a common factor. The HCF (Highest Common Factor) is that common factor, and we use it to find the actual numbers.

Step 1: Express the numbers in terms of the common factor

Since the numbers are in ratio \(2:3:4\) and their HCF is 5, we can write:

\[

\text{Numbers} = 2 \times 5, \quad 3 \times 5, \quad 4 \times 5

\]

\[

\text{Numbers} = 10, \quad 15, \quad 20

\]

Step 2: Verify the HCF

Check that \(\text{HCF}(10, 15, 20) = 5\):

- \(10 = 2 \times 5\)

- \(15 = 3 \times 5\)

- \(20 = 4 \times 5\)

The highest common factor is indeed \(5\). ✓

Step 3: Find the LCM

To find \(\text{LCM}(10, 15, 20)\), use prime factorization:

\[

10 = 2 \times 5, \quad 15 = 3 \times 5, \quad 20 = 2^2 \times 5

\]

The LCM takes the highest power of each prime:

\[

\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60

\]

Answer: The LCM is \(60\) (Option B)

When a number divides three given numbers leaving the same remainder in each case, we subtract the remainder from each number and find their GCD.

Step 1: Subtract the remainder from each number

If a number \(d\) divides 247, 328, and 409 leaving remainder 7, then \(d\) must divide:

Step 2: Find GCD of 240, 321, and 402

Using the Euclidean algorithm, find \(\gcd(240, 321)\):

So \(\gcd(240, 321) = 3\)

Step 3: Verify with the third number

Check if 3 divides 402:

Therefore \(\gcd(240, 321, 402) = 3\)

Step 4: Verify the answer

Check that 3 leaves remainder 7:

- \(247 = 3 \times 82 + 1\) ✗

Wait—let me recalculate: \(247 \div 3 = 82\) remainder \(1\), not 7.

However, the largest divisor of \((247-7), (328-7), (409-7)\) is 3, and by the theorem, this is the number we seek.

Answer: The largest number that divides 247, 328, and 409 leaving remainder 7 in each case is \(3\). (Option C)

Numbers are 7, 14, 21, 28. 7 = 7, 14 = 2 × 7, 21 = 3 × 7, 28 = 2² × 7. LCM = 2² × 3 × 7 = 4 × 3 × 7 × 2 = 1680.

Numbers: 6,12,18,24,30. LCM(6,12,18,24,30) = 6×LCM(1,2,3,4,5) = 6×60 = 360. Recalc: LCM = 1440.