Find the least number which when divided by 12, 18, 24 and 30 leaves 4 as remainder in each case but when divided by 7 leaves no remainder.

Options

A. 634
B. 366
C. 364
D. 384

Answer

To solve the problem of finding the least number which, when divided by 12, 18, 24, and 30 leaves a remainder of 4, and when divided by 7 leaves no remainder, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of 12, 18, 24, and 30

To find the LCM, we can use the prime factorization method:

- **12** = 2² × 3¹
- **18** = 2¹ × 3²
- **24** = 2³ × 3¹
- **30** = 2¹ × 3¹ × 5¹

Now, we take the highest power of each prime factor:

- For 2: the highest power is 2³ (from 24)
- For 3: the highest power is 3² (from 18)
- For 5: the highest power is 5¹ (from 30)

So, the LCM is calculated as:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360 \]

### Step 2: Adjust for the Remainder

Since we need a number that leaves a remainder of 4 when divided by 12, 18, 24, and 30, we can express our number \( N \) in the form:
\[ N = 360k + 4 \]
where \( k \) is a non-negative integer.

### Step 3: Ensure Divisibility by 7

We also need \( N \) to be divisible by 7:
\[ 360k + 4 \equiv 0 \ (\text{mod} \ 7) \]

First, we find \( 360 \mod 7 \):
\[ 360 \div 7 = 51 \quad \text{(remainder 3)} \]
So, \( 360 \equiv 3 \ (\text{mod} \ 7) \).

Now substituting this into our equation:
\[ 3k + 4 \equiv 0 \ (\text{mod} \ 7) \]
This simplifies to:
\[ 3k \equiv -4 \equiv 3 \ (\text{mod} \ 7) \]
Now we can add 7 to -4 to get a positive equivalent:
\[ 3k \equiv 3 \ (\text{mod} \ 7) \]

### Step 4: Solve for \( k \)

To solve for \( k \):
\[ k \equiv 1 \ (\text{mod} \ 7) \]
This means \( k \) can be expressed as:
\[ k = 7m + 1 \]
for some integer \( m \).

### Step 5: Substitute Back to Find \( N \)

Substituting \( k \) back into the equation for \( N \):
\[ N = 360(7m + 1) + 4 = 2520m + 360 + 4 = 2520m + 364 \]

### Step 6: Find the Least Value of \( N \)

To find the least number, we set \( m = 0 \):
\[ N = 364 \]

### Conclusion

Thus, the least number which when divided by 12, 18, 24, and 30 leaves a remainder of 4, and when divided by 7 leaves no remainder is:
\[ \boxed{364} \]

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