The least number which when divided by 35 leaves a remainder 25, when divided by 45 leaves the remainder 35 and when divided by 55 leaves 45 is

Options

A. 3465
B. 3645
C. 3655
D. 3455

Answer

To find the least number that meets the given conditions, we can follow these steps:

### Step 1: Understand the Conditions
We need to find a number \( N \) such that:
- \( N \mod 35 = 25 \)
- \( N \mod 45 = 35 \)
- \( N \mod 55 = 45 \)

### Step 2: Rewrite the Conditions
From the conditions, we can express \( N \) in terms of the divisors and remainders:
- \( N = 35k + 25 \) for some integer \( k \)
- \( N = 45m + 35 \) for some integer \( m \)
- \( N = 55n + 45 \) for some integer \( n \)

### Step 3: Find the Differences
We can find the difference between each divisor and its corresponding remainder:
- For 35: \( 35 - 25 = 10 \)
- For 45: \( 45 - 35 = 10 \)
- For 55: \( 55 - 45 = 10 \)

This shows that the common difference is 10.

### Step 4: Calculate the LCM of the Divisors
Next, we need to find the least common multiple (LCM) of the divisors 35, 45, and 55.

- The prime factorization of each number is:
- \( 35 = 5 \times 7 \)
- \( 45 = 5 \times 3^2 \)
- \( 55 = 5 \times 11 \)

- The LCM is found by taking the highest power of each prime:
- \( LCM = 3^2 \times 5^1 \times 7^1 \times 11^1 \)

Calculating this:
- \( LCM = 9 \times 5 \times 7 \times 11 \)
- \( 9 \times 5 = 45 \)
- \( 45 \times 7 = 315 \)
- \( 315 \times 11 = 3465 \)

So, \( LCM(35, 45, 55) = 3465 \).

### Step 5: Find the Least Number
Now, we need to subtract the common difference (10) from the LCM to find the least number:
- \( N = 3465 - 10 = 3455 \)

### Conclusion
Thus, the least number which meets all the conditions is **3455**.

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