Finding the Least Number to Ensure Divisibility

The problem asks for the smallest number that, when added to 478, makes the resulting sum exactly divisible by 5, 6, and 12. For a number to be exactly divisible by 5, 6, and 12 simultaneously, it must be a common multiple of these numbers. We are looking for the smallest such addition, which means the resulting number must be the smallest multiple of 5, 6, and 12 that is greater than or equal to 478. The smallest common multiple of a set of numbers is their Least Common Multiple (LCM).

Calculating the LCM of 5, 6, and 12

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

• Prime factorization of 5: $5^1$
• Prime factorization of 6: $2^1 \times 3^1$
• Prime factorization of 12: $2^2 \times 3^1$

The LCM is found by taking the highest power of all prime factors that appear in any of the numbers:

LCM(5, 6, 12) = $2^{\text{max}(0,1,2)} \times 3^{\text{max}(0,1,1)} \times 5^{\text{max}(1,0,0)}$

LCM(5, 6, 12) = $2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$

So, the resulting number must be a multiple of 60.

Finding the Next Multiple of 60 Greater than 478

We need to find the smallest multiple of 60 that is greater than or equal to 478. We can divide 478 by 60 to see where 478 falls in relation to multiples of 60.

Dividing 478 by 60:

$478 \div 60$

We can estimate how many times 60 goes into 478. $60 \times 7 = 420$ and $60 \times 8 = 480$.

So, 478 is between the 7th multiple (420) and the 8th multiple (480) of 60.

Using division with remainder:

$478 = 60 \times 7 + 58$

The remainder is 58. This means 478 is 58 more than a multiple of 60 (420). To reach the next multiple of 60 (480), we need to add the difference between the next multiple (480) and the current number (478).

Number to add = Next multiple of 60 - 478

Number to add = $480 - 478 = 2$

Therefore, adding 2 to 478 will result in 480, which is exactly divisible by 60 (and thus by 5, 6, and 12).

Verification

If we add 2 to 478, we get $478 + 2 = 480$.

• Is 480 divisible by 5? Yes, $480 \div 5 = 96$.
• Is 480 divisible by 6? Yes, $480 \div 6 = 80$.
• Is 480 divisible by 12? Yes, $480 \div 12 = 40$.

Since 480 is divisible by 5, 6, and 12, and it is the smallest multiple of their LCM (60) that is greater than 478, the least number to be added is indeed 2.

Revision Table: Divisibility and LCM Concepts

Additional Information: Finding LCM Using Prime Factorization

The prime factorization method for finding the LCM is a systematic way to ensure all necessary prime factors are included with their highest required powers.

Steps:

1. Find the prime factorization of each number.
2. List all unique prime factors from all the factorizations.
3. For each unique prime factor, find the highest power it appears with in any of the factorizations.
4. Multiply these highest powers together to get the LCM.

Example (LCM of 5, 6, 12):

• 5 = $5^1$
• 6 = $2^1 \times 3^1$
• 12 = $2^2 \times 3^1$

Unique prime factors are 2, 3, and 5.

• Highest power of 2: $2^2$ (from 12)
• Highest power of 3: $3^1$ (from 6 and 12)
• Highest power of 5: $5^1$ (from 5)

LCM = $2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$.