When you divide ♥ by 9, you get ⬛ groups of 9 with a remainder 4.

So ♥ = ⬛ x 9 + 4


Now 9 is divisible by 3. (9 ÷ 3 = 3)

So all those groups of 9 can be divided by 3. Which means you can create exact groups of 3 out from them.


The remainder of 4 however, is not divisible by 3. You can only make 1 group of 3 from it, and you will still have a remainder of 1.


So we need to add some number to this 4 such that it becomes divisible by 3.


We can add 2.

4 + 2 = 6
6 ÷ 3 = 2


So ♥ + 2 = ⬛ x 9 + 4 + 2

= (⬛ x 3 x 3 ) + 6

= (⬛ x 3) x 3 + 2 x 3

We have (3 x ⬛) groups of 3 plus 2 groups of 3.


Total we have (3 x ⬛ + 2) groups of 3

You can any number to that 4 , as long as you get a result that is a multiple of 3.

Eg.

4 + 8 = 12
12 ÷ 3 = 4

Or

4 + 5 = 9
9 ÷ 3 = 3



Thing to take note :

① If 2 or more numbers add up to a bigger number, and all the numbers are divisible by some other number n , then that bigger number is also divisible by n.

② If 2 or more numbers have a difference and all are divisible by some number n , then that difference is also divisible by n.

Dividing a number by 3 can be seen in two perspectives :

① Dividing that number into 3 equal parts.
② Dividing that number into groups of 3 , and seeing how many groups you can get.


Example :


15 = 3 x 5
So 15 ÷ 3 = 5

Perspective ① : 15 can be divided into 3 equal parts, where each part is 5.

Perspective ② : 15 can be divided into groups of 3, whereby you get 5 groups.

Thank you