Question

1. A certain integer "a" can be written in the form a = 4q+r. What are the possible
values of r when a is divided by 4?
(CN)​

Answer 1

Answer:

MATHS

Asked on December 20, 2019 byShivu Gabbar

Show that any odd positive integer is of the form 4q+1, or 4q+3.

ANSWER

We have

Any positive integer is of the form 4q+1or4q+3

As per Euclid’s Division lemma.

If a and b are two positive integers, then,

a=bq+r

Where 0≤r<b.

Let positive integers be a.and b=4

Hence,a=bq+r

Where, (0≤r<4)

R is an integer greater than or equal to 0 and less than 4

Hence, r can be either 0,1,2and3

Now, If r=1

Then, our be equation is becomes

a=bq+r

a=4q+1

This will always be odd integer.

Now, If r=3

Then, our be equation is becomes

a=bq+r

a=4q+3

This will always be odd integer.

Hence proved.

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Given, a = 4q+r

According to Euclid's division lemma,

a = bq + r where 'r' lies between 0 & b including 0

here b = 4

=> possible values of r = 0 , 1 , 2 , 3

If r= 0 , then a = 4q + 0 = 4q

If r= 1 , then a = 4q + 1

If r= 2 , then a = 4q + 2

If r= 3 , then a = 4q + 3

Also given, a is divided by 4

so from the above, 4q is only divisible by 4

.•. possible values of r when a is divided by 4 are

r = 0