Divisibility

If you have read the divisibility rules page, we now extend the numbers to include all integers. Also, we will give you the formal definition for divisibility:

An integer, n, is divisible by another integer, m, (m ¹ 0) if and only if n = mq for some integer q, and we write m|n. If no such q exists, we say n is not divisible by m and we write mšn.

For example, 6 is divisible by 2 since 6 = 2(3). Here n = 6, m = 2 and q = 3. 5 is not divisible by 2 since no integer in the world multiplied by 2 will yield 5.

Of course, there are other ways to say one integer is divisible by another. If n is divisible by m, we also can say:

·  n is a multiple of m;

·  m is a factor of n;

·  m is a divisor of n;

·  m divides (into) n; and

·  m goes into n.

 

It should be noted here that the term “factor” is used mainly in elementary school and secondary school. For example, the greatest common factor (GCF) of 16 and 24 is 8 and the GCF of 16x2y and 24xy2 is 8xy. In number theory, however, we tend to use exclusively the term “divisor.” For example, the greatest common divisor of 16 and 24 is 8, denoted by gcd(16, 24) = 8, or simply (16, 24) = 8. On the other hand, the term “multiple” is still used here. For example, the least common multiple of 16 and 24 is 48, denoted by lcm[16, 24] = 48, or simply [16, 24] = 48.

 

Now I am going to give you the divisibility rules for 7 and 11:

 

Divisibility rules for 7

For a positive integer, n, subtract double the units digit of n from the integer obtained from n by removing the units digits. If the result is divisible by 7, then n is divisible by 7. You can repeat this process as many times as you want (i.e., until you can see the result is divisible by 7).

For example, 5,523 is divisible by 7 because 552 – 2(3) = 546 and 54 – 2(6) = 35 and 35 is obviously divisible by 7. 

 

Divisibility rules for 11

A positive integer, n, is divisible  by 11 if and only if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by 11.

For example, 4,675 is divisible by 11 since (6 + 5) – (4 + 7) = 0 and 0 is divisible by 11.