The least number which when divided by 16, 18, 20 and 25 leaves 4 as remainder in each cae but when divided by 7 leaves no remainder is

Options

A. 17004
B. 18000
C. 18002
D. 18004

Answer

To solve the problem step by step, we need to find the least number that meets the given conditions. Let's break it down:

### Step 1: Understand the Problem
We need to find a number that:
- When divided by 16, 18, 20, and 25 leaves a remainder of 4.
- When divided by 7 leaves no remainder.

### Step 2: Adjust the Condition for Remainders
If a number leaves a remainder of 4 when divided by 16, 18, 20, and 25, we can express this number as:
\[ N = k \cdot \text{LCM}(16, 18, 20, 25) + 4 \]
where \( k \) is some integer.

### Step 3: Find the LCM of 16, 18, 20, and 25
To find the least common multiple (LCM), we can use the prime factorization method:
- \( 16 = 2^4 \)
- \( 18 = 2 \cdot 3^2 \)
- \( 20 = 2^2 \cdot 5 \)
- \( 25 = 5^2 \)

Now, take the highest power of each prime:
- For 2: \( 2^4 \)
- For 3: \( 3^2 \)
- For 5: \( 5^2 \)

Thus,
\[ \text{LCM}(16, 18, 20, 25) = 2^4 \cdot 3^2 \cdot 5^2 \]

Calculating this:
\[ 2^4 = 16, \quad 3^2 = 9, \quad 5^2 = 25 \]
\[ \text{LCM} = 16 \cdot 9 \cdot 25 \]
Calculating step by step:
\[ 16 \cdot 9 = 144 \]
\[ 144 \cdot 25 = 3600 \]
So, \( \text{LCM}(16, 18, 20, 25) = 3600 \).

### Step 4: Express the Number
Now, we can express our number \( N \) as:
\[ N = 3600k + 4 \]

### Step 5: Check the Condition for Divisibility by 7
We need \( N \) to be divisible by 7:
\[ 3600k + 4 \equiv 0 \mod 7 \]
Calculating \( 3600 \mod 7 \):
\[ 3600 \div 7 = 514 \quad \text{(remainder 2)} \]
So, \( 3600 \equiv 2 \mod 7 \).

Now substituting:
\[ 2k + 4 \equiv 0 \mod 7 \]
This simplifies to:
\[ 2k \equiv -4 \mod 7 \quad \Rightarrow \quad 2k \equiv 3 \mod 7 \]

### Step 6: Solve for \( k \)
To solve \( 2k \equiv 3 \mod 7 \), we can find the multiplicative inverse of 2 modulo 7. The inverse is 4 since:
\[ 2 \cdot 4 \equiv 1 \mod 7 \]
Multiplying both sides of \( 2k \equiv 3 \) by 4:
\[ k \equiv 12 \mod 7 \quad \Rightarrow \quad k \equiv 5 \mod 7 \]

### Step 7: Find the Least Value of \( k \)
The smallest non-negative integer satisfying \( k \equiv 5 \mod 7 \) is \( k = 5 \).

### Step 8: Calculate \( N \)
Substituting \( k = 5 \) into the equation for \( N \):
\[ N = 3600 \cdot 5 + 4 = 18000 + 4 = 18004 \]

### Conclusion
The least number that satisfies all the conditions is:
\[ \boxed{18004} \]