Question

21
23. Justify:
(a) If x leaves a remainder of 1 when divided by 5, then x also leaves a remainder of 1 when divided by 5.
(b) If x is an even natural number then x is also an even number.
(c)
If x is an odd number then x is also an odd number.
(d) If x is a negative integer then x is also a negative integer.
(e) If o ends in 4 then dends in 8.
(f) If d'ends in 9 then d' ends in 7.
(g) If d’ends in 6 then aends in 6.​

Answer 1

Answer:

If  and  are positive integers, there exist unique integers  and , called the quotient and remainder, respectively, such that  and .

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since .

Notice that  means that remainder is a non-negative integer and always less than divisor.

This formula can also be written as .

Properties

When  is divided by  the remainder is 0 if  is a multiple of .

For example, 12 divided by 3 yields the remainder of 0 since 12 is a multiple of 3 and .

When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.

For example, 7 divided by 11 has the quotient 0 and the remainder 7 since 

The possible remainders when positive integer  is divided by positive integer  can range from 0 to .

For example, possible remainders when positive integer  is divided by 5 can range from 0 (when y is a multiple of 5) to 4 (when y is one less than a multiple of 5).

If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.

For example, 123 divided by 10 has the remainder 3 and 123 divided by 100 has the remainder of 23.