Rectangles of 12
Take 12 counters and arrange them in a full rectangle, with every row the same length and no gaps. 2 rows of 6 works. The counters share into 2 rows exactly, and into rows of 6 exactly, so 2 and 6 both divide 12.
A number that divides 12 exactly is called a factor of 12. So 2 and 6 are factors of 12.
2 rows of 6 make 12, so 2 and 6 are factors of 12.
The same 12 counters also make 3 rows of 4. So 3 and 4 are factors of 12 as well: 12 ÷ 3 = 4 and 12 ÷ 4 = 3, with nothing left over.
3 rows of 4 make 12, so 3 and 4 are factors of 12.
1 and the number itself
One long row is a rectangle too: 1 row of 12. So 1 and 12 divide 12, and they are factors of 12.
This is true of every whole number. Any number of counters can stand in one row, so 1 and the number itself are always factors.
One row of 12: 1 and 12 are factors of 12.
All the factors of 12
The rectangles found 1, 2, 3, 4, 6 and 12, so 12 has six factors. No other number divides 12 exactly. 5 does not: rows of 5 use 10 counters and leave 2 over. 7, 8, 9, 10 and 11 each leave a remainder as well.
A factor of 12 is never bigger than 12, because a number bigger than 12 cannot make even one row. So the list of factors has an end.
The factors of 12 are ringed. Each number that is not ringed leaves a remainder when it divides 12.
Now 24
24 counters make more rectangles. 3 rows of 8 is one of them, so 3 and 8 are factors of 24. 4 rows of 6 is another, so 4 and 6 are factors of 24.
multiplication is commutative: 3 × 5 = 5 × 3
Make 24 as a rectangle, then find another factor pair
Drag the corner to change the number of rows and columns. Each rectangle of 24 dots gives two factors of 24.
Two slips
Leaving out 1 and the number itself is the most common slip. They are factors like any others: 1 × 12 = 12.
A number that leaves a remainder is not a factor, however close it comes. 12 ÷ 5 = 2 remainder 2, so 5 is not a factor of 12.