The numbers 400, 536 and 646, when divided by a number N, give the same remainder of 22, 23 and 25 respectively. Find the greatest such number N.

Answer

To find the greatest number \( N \) such that the numbers 400, 536, and 646 give the remainders 22, 23, and 25 respectively when divided by \( N \), we can follow these steps:

### Step 1: Set Up the Equations
From the problem, we can express the conditions mathematically:
- \( 400 \equiv 22 \mod N \)
- \( 536 \equiv 23 \mod N \)
- \( 646 \equiv 25 \mod N \)

This implies:
- \( 400 - 22 = 378 \) is divisible by \( N \)
- \( 536 - 23 = 513 \) is divisible by \( N \)
- \( 646 - 25 = 621 \) is divisible by \( N \)

### Step 2: Calculate the Values
Now, we can write:
- \( 378 \)
- \( 513 \)
- \( 621 \)

### Step 3: Find the HCF
To find the greatest number \( N \), we need to calculate the highest common factor (HCF) of these three numbers:
- \( N = \text{HCF}(378, 513, 621) \)

### Step 4: Factor Each Number
Now, we will factor each of these numbers:
1. **Factor 378**:
- \( 378 = 2 \times 3^3 \times 7 \)

2. **Factor 513**:
- \( 513 = 3^3 \times 19 \)

3. **Factor 621**:
- \( 621 = 3^3 \times 23 \)

### Step 5: Identify Common Factors
Now, we look for the common factors in the factorizations:
- The common factor among \( 378 \), \( 513 \), and \( 621 \) is \( 3^3 = 27 \).

### Step 6: Conclusion
Thus, the greatest number \( N \) that satisfies the conditions is:
\[ N = 27 \]