Euclid Division Lemma


 
 
Concept Explanation
 

Euclid Division Lemma

Euclid Division Lemma:

If a is an integer and b is a positive integer then there exist two integers q and r such that

                  large a=bq+r;;where;;0leq r< b

q and r are called quotient and remainder respectively when a is divided by b.

For instance, if a = -86, b = 5 then -86 =5(-18) + 4

Thus, q = -18  and r = 4

Note that q and r in the above statement are unique.

Example: Show that any positive even integer is of the form 8q, 8q +2, 8q + 4, 8q + 6, where q is some integer.

Solutions:

Let a and b = 8 be two positive integers where a is even.

Applying division lemma a = 8q + r where 0 leq r <8

So, r can take any of the values 0, 1, 2, 3, 4, 5, 6, 7

Therefore, a = 8q, 8q + 1, 8q + 2, 8q + 3, 8q + 4, 8q +5, 8q + 6,8q + 7, 8q + 8 

Since, a is even.

Therefore, a can take values 8q,  8q + 2, 8q + 4, 8q + 8 since they can expressed as multiples of 2.

So, a will not take value 8q + 1, 8q + 3, 8q + 5, 8q + 7. as they all represent odd numbers.

Sample Questions
(More Questions for each concept available in Login)
Question : 1

On applying division lemma to 17 and, 7, You get 17 = a X 7 + 3. What is the value of a?

Right Option : B
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Explanation
Question : 2

On applying division Lemma to 25 and X, You get 25 = 8X + 1 . What is the value of x?

Right Option : C
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Explanation
Question : 3

On applying division lemma to 36 and 5, you get

Right Option : A
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Explanation
 
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