what is Euclid division lemma?
Answers
Answer
Euclid division lemma says that
a = b* q + r (where a and b both are natural numbers and 0 < = r < b)
EUCLID'S DIVISION LEMMA - DEFINITION
Let a and b be any two positive integers. Then there exist unique integers q and r such that
a=bq+r,0rb.
If b∣a, then r=0.
Otherwise, r satisfies the stronger inequality 0rb.
HCF USING EUCLID'S DIVISON - EXAMPLE
If a and b are positive integers such that a=bq+r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.
Example: Find HCF of 420 and 130.
Since 420>130 we apply the division lemma to 420 and 130 to get ,Since 300 , we apply the division lemma to 130 and 30 to get
130=30×4+10
420=130×3+30
Since 100 , we apply the division lemma to 30 and 10 to get
30=10×3+0
The remainder has now become zero, so our procedure stops. Since the divisor at this step is 10, the HCF of 420 and 130 is 10.
MULTIPLICATIVE INVERSE - RESULT
Multiplicative inverse of any irrational number b
=0 is equal to
b
1
denoted by b
−1
such that : b×b
−1
=1
. Example:
3
1 is the multiplicative inverse of
3