The HCF and LCM of two numbers are 27 and 405 respectively. If one of the numbers is divided by 7, then remainder is 4. What is the other number?

Options

A. 90
B. 108
C. 135
D. 85

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript and apply the necessary mathematical principles.

### Step 1: Understand the given information
We know that:
- The HCF (Highest Common Factor) of two numbers is 27.
- The LCM (Lowest Common Multiple) of the same two numbers is 405.
- One of the numbers, when divided by 7, leaves a remainder of 4.

### Step 2: Use the Division Theorem
According to the division theorem, we can express the number in terms of its quotient and remainder. If we let the number be \( x \), then:
\[ x = 7 \cdot q + r \]
Where:
- \( q \) is the quotient,
- \( r \) is the remainder (which is 4 in this case).

So we have:
\[ x = 7q + 4 \]

### Step 3: Find the value of \( x \)
Since we know that when \( x \) is divided by 7, the remainder is 4, we can express \( x \) in a way that fits this condition. To find a suitable \( q \), we can start with \( q = 11 \) (as a reasonable integer choice):
\[ x = 7 \cdot 11 + 4 = 77 + 4 = 81 \]
Thus, one of the numbers is 81.

### Step 4: Use the relationship between HCF, LCM, and the two numbers
We know that:
\[ \text{HCF} \times \text{LCM} = \text{Number 1} \times \text{Number 2} \]
Substituting the known values:
\[ 27 \times 405 = 81 \times \text{Number 2} \]

### Step 5: Calculate the product of HCF and LCM
Calculating \( 27 \times 405 \):
\[ 27 \times 405 = 10935 \]

### Step 6: Solve for the other number
Now we can set up the equation:
\[ 10935 = 81 \times \text{Number 2} \]
To find the other number, we divide both sides by 81:
\[ \text{Number 2} = \frac{10935}{81} \]

### Step 7: Perform the division
Calculating \( \frac{10935}{81} \):
\[ \text{Number 2} = 135 \]

### Conclusion
The other number is **135**.

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