The H.C.F. and L.C.M. of two 2 - digit numbers are 16 and 480 re spectivley. The numbers are :

Options

A. 40, 48
B. 60, 72
C. 64, 80
D. 80, 96

Answer

To find the two-digit numbers whose H.C.F. is 16 and L.C.M. is 480, we can use the relationship between H.C.F., L.C.M., and the two numbers.

**Step 1: Use the relationship between H.C.F., L.C.M., and the two numbers.**

The relationship states that:
\[ \text{H.C.F.} \times \text{L.C.M.} = \text{First Number} \times \text{Second Number} \]

Given:
- H.C.F. = 16
- L.C.M. = 480

So, we can write:
\[ 16 \times 480 = \text{First Number} \times \text{Second Number} \]

**Step 2: Calculate the product of the two numbers.**

Calculating the left side:
\[ 16 \times 480 = 7680 \]

Thus, we have:
\[ \text{First Number} \times \text{Second Number} = 7680 \]

**Step 3: Let the two numbers be \( 16a \) and \( 16b \).**

Since the H.C.F. is 16, we can express the two numbers as:
- First Number = \( 16a \)
- Second Number = \( 16b \)

where \( a \) and \( b \) are co-prime integers (i.e., their H.C.F. is 1).

**Step 4: Substitute into the product equation.**

Substituting into the product equation:
\[ (16a) \times (16b) = 7680 \]
\[ 256ab = 7680 \]

**Step 5: Solve for \( ab \).**

Dividing both sides by 256:
\[ ab = \frac{7680}{256} = 30 \]

**Step 6: Find pairs of co-prime integers \( (a, b) \) such that \( ab = 30 \).**

The pairs of integers that multiply to 30 are:
- (1, 30)
- (2, 15)
- (3, 10)
- (5, 6)

Among these pairs, we need to find co-prime pairs:
- (1, 30) - co-prime
- (2, 15) - co-prime
- (3, 10) - co-prime
- (5, 6) - co-prime

**Step 7: Calculate the two-digit numbers.**

Now, we can calculate the two-digit numbers for each co-prime pair:
1. For \( (1, 30) \):
- First Number = \( 16 \times 1 = 16 \) (not a two-digit number)
- Second Number = \( 16 \times 30 = 480 \) (not a two-digit number)

2. For \( (2, 15) \):
- First Number = \( 16 \times 2 = 32 \)
- Second Number = \( 16 \times 15 = 240 \) (not a two-digit number)

3. For \( (3, 10) \):
- First Number = \( 16 \times 3 = 48 \)
- Second Number = \( 16 \times 10 = 160 \) (not a two-digit number)

4. For \( (5, 6) \):
- First Number = \( 16 \times 5 = 80 \)
- Second Number = \( 16 \times 6 = 96 \) (not a two-digit number)

The valid two-digit numbers found are:
- 48 and 80

**Final Answer: The two-digit numbers are 48 and 80.**

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