Page 5 - Matematica_Mathematics - Teorema impartirii cu rest_The Long division theorem

Mathematically speaking, 13 : 4 = 3 (remainder 1)

That means 3 × 4 + 1 = 13, since the total number of candies is equal to 3 × 4, that is 12 (divided
candies) + 1 (the candy that was left extra).
In this division, 13 is the dividend (“deîmpărțitul”), 4 is divider (“împărțitorul”), 3 is quotient (“câtul”)
and 1 is the remainder (“restul”).
You are lucky also to have 14 or 15 candies. Basically, in each
situation you will share 12 candies, and the rest will remain with you.
14 : 4 = 3 (remainder 2), that is 4 × 3 + 2 = 14
15 : 4 = 3 (remainder 3), that is 4 × 3 + 3 = 15

If you have 16 candies and give 3 to every friend, you have left 4, that is exactly one more for each
friend. This time, 16 : 4 = 4 (remainder 0).

Practice

2. Find the quotient and the remainder of the divisions 37: 4; 52: 6; 99: 8; 100: 9, then use the long
division theorem to make the sample.

Use pattern: 46: 6 = 7 (remainder 4) 7 × 6 + 4 = 46

We can deduce, therefore, that when we divide a natural number a to another natural number b, we
get a quotient (c) and a remainder (r). The remainder may be equal to or different from 0, but it is always less
than the divisor.
The divider is always different from 0 and is bigger than the remainder.
Dividing to 0 does not make sense and this is easy to prove.

Let’s take a natural number a, different from 0, which we will divide by 0. We will have a : 0 = b, which
means that b × 0 = a. But since the product of a number with 0 is 0, it means that a = 0, which contradicts
the statement I made at the beginning. We will conclude that dividing by 0 does not make sense, that is why
the divider cannot be 0.