Question

List all elements of each set. (Note that sometimes you will have to put three dots at the end, because a set is infinite.)
E={all possible remainders from division of a natural number by 6}

Answer 1

Whenever we divide a number by other number, the 'left-over' amount is known as the Remainder.

Set of natural numbers N is defined as following:

$N = \{1,2,3,4,5,6,7,8,9,10 ...\}$

If we divide any natural number by 6 taken from the set N, the possible remainders can be 0, 1, 2, 3, 4, 5.

1. Consider the following set A (a subset from N):

$A = \{6, 12, 18, 24, 30, 36, ...\}$

When we divide any number from the set A by 6, remainder will be 0.

2. Consider the following set B (a subset from N):

$B = \{1, 7, 13, 19, 25, 31, 37, ...\}$

When we divide any number from the set B by 6, remainder will be 1.

3. Consider the following set C (a subset from N):

$C = \{2, 8, 14, 20, 26, 32, 38, ...\}$

When we divide any number from the set C by 6, remainder will be 2.

4. Consider the following set D (a subset from N):

$D = \{3, 9, 15, 21, 27, 33, 39, ...\}$

When we divide any number from the set D by 6, remainder will be 3.

5. Consider the following set F (a subset from N):

$F = \{4, 10, 16, 22, 28, 34, 40, ...\}$

When we divide any number from the set F by 6, remainder will be 4.

6. Consider the following set G (a subset from N):

$G = \{5, 11, 17, 23, 29, 35, 41, ...\}$

When we divide any number from the set G by 6, remainder will be 5.

So, E, the set of all possible remainders when we divide any natural number by 6 is:

$E = \{0,1,2,3,4,5\}$