THINK & Discuss
= bq+r?
If r=0, then what is the relationship between a, b and q in a=bq, 'a'+r?
Answers
Answer:
If r=0, then what is the relationship between a,b and q in a=bq+r.
Answer:
Hint: We will substitute the value of r=0 given in the question to the equation given in the question and find the relationship between a,b and q in a=bq+r . Euclid’s Division Lemma states that if two positive integers a and b, then there exist two unique integers q and r such that a=bq+r where 0⩽r<b.
Complete step-by-step solution:
The given value of r is 0 in the question.
The given equation is a=bq+r .
This equation is known as Euclid’s division lemma.
Now we will substitute the value of r=0 in the equation, we will get,
a=bq+0
Adding any number with 0 gives us the same number unchanged.
Therefore, a=bq
Therefore, the relationship between a,b and q from the equation after substituting r=0 is a=bq.
Since, b divides a , we can conclude that b is a factor of a.
Note: According to Euclid’s division lemma:
a=bq+r where a is dividend, b is divisor, q is quotient and r is the remainder.
Here, remainder is always smaller than the divisor and it is equal to or more than 0.
Therefore, 0⩽r<b , when r=0 , a is completely divisible by b.