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✨Example of Euclid division Lemma ✨
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ANSWER
EUCLID DIVISION LEMMA
Given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Euclid’s division algorithm is based on this lemma.
find the HCF of the integers 455 and 42. We start with the larger integer, that is,
455. Then we use Euclid’s lemma to get
455 = 42 × 10 + 35
Now consider the divisor 42 and the remainder 35, and apply the division lemma
to get
42 = 35 × 1 + 7
Now consider the divisor 35 and the remainder 7, and apply the division lemma
to get
35 = 7 × 5 + 0
the remainder has become zero.
We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7.
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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. 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.
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. 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.
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. 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.
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. 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.