Properties of Division on Integers:
Property 1:
If a and b are integers then (a/b) is not necessarily an integer.
Verification: We have,
1) 14 and 3 are both integers, but (14/3) is not an integer.
2) 17 and 6 are both integers, but (17/6) is not an integer.
Property 2:
If a is an integer and a is not equal to 0 then (a/a) = 1.
Verification: We have,
1) (4/4) = 1.
2) ((-5)/(-5)) = 1.
Property 3:
If a is an integer, then (a/1) = a.
Verification: We have,
1) (6/1) = 6.
2) ((-17)/1) = -17.
Property 4:
If a is a non zero integer, then (0/a) = 0, but (a/0) is not meaningful. Division by Zero is undefined.
Verification: We have,
1) (0/2) = 0.
2) (0/(-9)) = 0.
3) (6/0) is not meaningful.
(a/0) is not meaningful
Property 5:
If a, b, c are any three integers then (a/b)/c is not equal to a/(b/c), unless c= 1.
Verification:
Let a = 8, b = 4 and c = 2. Then,
(a/b)/c = (8/4)/2 = (2/2) = 1.
a/(b/c) = 8/(4/2) =(8/2) = 4.
Therefore, (a/b)/c is not equal to a/(b/c).
If c = 1 then,
(a/b)/c = (8/4)/1 = (2/1) = 2.
a/(b/c) = 8/(4/1) =(8/4) = 2.
Therefore, In this case (a/b)/c is equal to a/(b/c).
Property 6:
If a, b, c are integers and a > b then
1) (a/b) > (b/c), if c is positive
2) (a/c) < (b/c), if c is negative
Verification:
1) 27 > 18, and 9 is positive, that implies 27/9 > 18/9
2) 27 > 18, and (-9) is negative, that implies 27/(-9) < 18/(-9)