Which of the following is the smallest positive integer which gives the remaninder 3 when divided with 4,6,8 and 10?

Options

A. 73
B. 103
C. 123
D. 234

Answer

To solve the problem of finding the smallest positive integer \( X \) that gives a remainder of 3 when divided by 4, 6, 8, and 10, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( X \) such that:
- \( X \mod 4 = 3 \)
- \( X \mod 6 = 3 \)
- \( X \mod 8 = 3 \)
- \( X \mod 10 = 3 \)

This means that \( X - 3 \) must be divisible by each of these numbers (4, 6, 8, and 10).

### Step 2: Set Up the Equation
Let:
\[ Y = X - 3 \]
Then, \( Y \) must satisfy:
- \( Y \mod 4 = 0 \)
- \( Y \mod 6 = 0 \)
- \( Y \mod 8 = 0 \)
- \( Y \mod 10 = 0 \)

This implies that \( Y \) is a common multiple of 4, 6, 8, and 10.

### Step 3: Find the Least Common Multiple (LCM)
To find \( Y \), we need to calculate the LCM of the numbers 4, 6, 8, and 10.

**Prime Factorization:**
- \( 4 = 2^2 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 8 = 2^3 \)
- \( 10 = 2^1 \times 5^1 \)

**LCM Calculation:**
The LCM is found by taking the highest power of each prime factor:
- For \( 2 \): highest power is \( 2^3 \) (from 8)
- For \( 3 \): highest power is \( 3^1 \) (from 6)
- For \( 5 \): highest power is \( 5^1 \) (from 10)

Thus, the LCM is:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]

Calculating this step-by-step:
1. \( 8 \times 3 = 24 \)
2. \( 24 \times 5 = 120 \)

So, \( \text{LCM}(4, 6, 8, 10) = 120 \).

### Step 4: Calculate \( Y \)
Since \( Y \) is the LCM, we have:
\[ Y = 120 \]

### Step 5: Find \( X \)
Now, recall that \( X = Y + 3 \):
\[ X = 120 + 3 = 123 \]

### Conclusion
The smallest positive integer \( X \) that gives a remainder of 3 when divided by 4, 6, 8, and 10 is:
\[ \boxed{123} \]