find the greatest number that will divide by 20,25,35 and 40 leaves the same remainder in each case.

Explanation

To find the greatest number that will divide 20, 25, 35, and 40 leaving the same remainder in each case, we can follow these steps:

### Step 1: Understand the problem
We need to find a number \( N \) such that when we divide 20, 25, 35, and 40 by \( N \), the remainder is the same for all four numbers.

### Step 2: Set up the equations
If \( N \) divides these numbers and leaves the same remainder \( r \), we can express this as:
- \( 20 = k_1N + r \)
- \( 25 = k_2N + r \)
- \( 35 = k_3N + r \)
- \( 40 = k_4N + r \)

Where \( k_1, k_2, k_3, k_4 \) are integers.

### Step 3: Subtract the equations
By subtracting the equations pairwise, we eliminate \( r \):
- \( 25 - 20 = (k_2 - k_1)N \) → \( 5 = (k_2 - k_1)N \)
- \( 35 - 25 = (k_3 - k_2)N \) → \( 10 = (k_3 - k_2)N \)
- \( 40 - 35 = (k_4 - k_3)N \) → \( 5 = (k_4 - k_3)N \)
- \( 40 - 20 = (k_4 - k_1)N \) → \( 20 = (k_4 - k_1)N \)

### Step 4: Find the differences
We can summarize the differences:
- \( 25 - 20 = 5 \)
- \( 35 - 25 = 10 \)
- \( 40 - 35 = 5 \)
- \( 40 - 20 = 20 \)

### Step 5: Calculate the HCF
Now, we need to find the highest common factor (HCF) of these differences:
- The differences are: 5, 10, 5, and 20.

### Step 6: Find the HCF
The HCF of 5, 10, and 20 is:
- Factors of 5: \( 1, 5 \)
- Factors of 10: \( 1, 2, 5, 10 \)
- Factors of 20: \( 1, 2, 4, 5, 10, 20 \)

The common factors are \( 1 \) and \( 5 \). Thus, the HCF is \( 5 \).

### Conclusion
The greatest number that will divide 20, 25, 35, and 40 leaving the same remainder is **5**.

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