Page 4 - Matematica_Mathematics - Divizibilitatea_Divisibility

6. Divisibility of natural numbers

As we have seen, in our lives numbers are very important. But
the world of numbers is really fascinating because there are a lot of
relationships between them. In fifth grade you will learn to differentiate
between mathematical operations and mathematical relationships.
Addition, subtraction, multiplication and division are not
mathematical relations. They are operations.
Relationship (“Relație”) means connection. For example, what is
the connection between 14 and 7? We know that 14 divides exactly by
7, that is, 14 is twice as large as 7.
Comparing numbers is a relationship and the symbol > means
that 14 is bigger than 7. In this way, we have established a relationship
between 7 and 14.

14 > 7
Divisibility is also a mathematical relationship. Its understanding
is based on the operations of multiplication and division.
PStarting from a concrete example, we will identify all natural
numbers less than 30 that divide exactly by 7, that is, those numbers
that by dividing by 7 give the remainder 0. The easiest thing is to start
from the multiplication table.

0 × 7 = 0
1 × 7 = 7
2 × 7 = 14
3 × 7 = 21
4 × 7 = 28

That means the numbers: 0, 7, 14, 21 and 28 they are divisible by 7, since each of them is the product of
a multiplication in which one of the factors is 7.
We can also establish that 56 is divisible by 7, since 56 = 7 × 8.

Practice
1. Find all two-digit numbers that are divisible by 9.

But 60, is it divisible by 7? Since there is no natural number multiplied by 7 to have the product 60
(7 × 8 = 56, and 7 × 9 = 63), we can safely say that 60 is not divisible by 7.
Generalizing, we say that a natural number a is divisible by the natural number b, if there is a natural
number c, so that a= b x c.