Finding Factors Using Rectangular Arrangements and Division

Imagine you have 12 desks to arrange in your classroom. How many different ways can you arrange them so that each row has the same number of desks and no desk is left unused? Each arrangement gives you a clue about the divisors (factors) of 12.

1. For 2 coins

Example:

• 1 × 2
• 2 × 1
• Factors of 2: 1, 2.

2. For 4 coins
 
Example:

• 1 × 4
• 2 × 2
• 4 × 1
• Factors of 4: 1, 2, 4.

3. For 6 coins

Example:

• 1 × 6
• 2 × 3
• 3 × 2
• 6 × 1
• Factors of 6: 1, 2, 3, 6.

If the remainder = 0, it's a factor.

1. On dividing 6 by 1, we get a quotient = 6 and a remainder = 0. 
   `1")"overline(6)"("6`
     - 6  
       0
   
2. On dividing 6 by 2, we get Quotient = 3 and Remainder = 0. 
   `2")"overline(6)"("3`
     - 6  
       0
3. On dividing 6 by 3, we get Quotient = 2 and Remainder = 0. 
   `3")"overline(6)"("2`
     - 6  
       0
4. On dividing 6 by 4, we get Quotient = 1 and Remainder = 2. 
   `4")"overline(6)"("1`
     - 6  
       2
5. On dividing 6 by 5, we get Quotient = 1 and Remainder = 1. 
   `5")"overline(6)"("1`
     - 5  
       1
6.  On dividing 6 by 6, we get Quotient = 1 and Remainder = 0. 
   `6")"overline(6)"("1`
     - 6  
       0

Here, we see that 1, 2, 3, and 6 are exact divisors of 6 and are called factors of 6. 
Therefore, factors of 6 = 1, 2, 3, 6

You want to put them in boxes so that each box has the same number of chocolates, without any leftover.

• 1 box of 12

• 2 boxes of 6

• 3 boxes of 4

• 4 boxes of 3

• 6 boxes of 2

• 12 boxes of 1

Each arrangement demonstrates a factor pair of 12.

• A factor divides a number exactly (no remainder).

• Rectangular arrangements represent all possible factor pairs.

• Factors always come in pairs (row × column).

• You can use division to check if a number is a factor.