In This Video We Learn Euclid's Division Theorem
Euclid's Division Theorem is combination of Euclid's Division Lemma and Euclid's Division Algorithm.
Euclid's Division Theorem State : for Given Positive Integer a and b. there exist unique integers q and r which satisfying the equation
a = q times b + r
where r is greater than and equal to 0 or smaller than b
Euclid's Division Algorithm: is a group or set of instruction which use to find the HCF of Two given positive integers.
it contain some steps which we have to follow to find the HCF of any two given positive integers.
To obtain the HCF of two positive integers, say c and d, with c is greater than d, follow
the steps below:
Step 1 Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = d multiply q + r, r is greater or equal to 0 or smaller than b
Step 2 If r = 0, d is the HCF of c and d. If r 0, apply the division lemma to d and r.
Step 3 Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
Example ; find the HCF of the integers 455 and 42 with the help of euclid's theorem or algorithm.
Solution:
Let us see how the algorithm works, through an example first. Suppose we need to 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
Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7.
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