Find the least number which should be added to 1456 so that it is divisible by 6,5 and 4 without leaving a remainder.

Options

A. 6
B. 61
C. 44
D. 16

Answer

To find the least number that should be added to 1456 so that it is divisible by 6, 5, and 4, we can follow these steps:

### Step 1: Find the LCM of 6, 5, and 4
To ensure that the number is divisible by 6, 5, and 4, we first need to find the least common multiple (LCM) of these numbers.

- The prime factorization of each number is:
- 6 = 2 × 3
- 5 = 5
- 4 = 2²

To find the LCM, we take the highest power of each prime number:
- For 2, the highest power is 2² (from 4).
- For 3, the highest power is 3¹ (from 6).
- For 5, the highest power is 5¹ (from 5).

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Determine the remainder when 1456 is divided by 60
Next, we need to find out how much 1456 exceeds the nearest multiple of 60. We do this by dividing 1456 by 60 and finding the remainder.

Perform the division:
\[ 1456 \div 60 = 24.2667 \]
The integer part of the quotient is 24. Now, we multiply it back by 60 to find the largest multiple of 60 less than 1456:
\[ 24 \times 60 = 1440 \]

Now, we find the remainder:
\[ 1456 - 1440 = 16 \]

### Step 3: Calculate the number to be added
To make 1456 divisible by 60, we need to add enough to reach the next multiple of 60. Since the remainder is 16, we need to add:
\[ 60 - 16 = 44 \]

### Final Answer
Thus, the least number that should be added to 1456 to make it divisible by 6, 5, and 4 is **44**.

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