Proof of Euclid division lemma class 10 for negative and positive integers

Proof of Euclid division lemma class 10 for negative and positive integers. Proof of Euclid's Division Lemma or Algorithm for positive and negative integers. You will learn proof of Euclid’s division lemma or algorithm by using simple division techniques you have already learnt in 2nd or 3rd grade. Given two integers a and b, with b not equal to zero, there exist unique integers q and r such that a=bq+r, where a is dividend, b is divisor, q is quotient and r is remainder. The remainder has unique properties; its value is greater than or equal to zero and less than divisor. The divisor must not be equal to zero; because Euclid’s division algorithm is not valid for zero divisor. Its only valid for integers. You may not be knowing that Euclid’s division algorithm can be extended for negative integers. I have taken example based on Euclid’s division lemma which is as follows:-
Euclid’s division algorithm (lemma) when divisor is negative integer and dividend is positive integer.
Euclid’s division algorithm (lemma) when divisor is negative integer and dividend is negative integer.
Euclid’s division algorithm (lemma) when divisor is positive integer and dividend is negative integer.
Euclid’s division algorithm (lemma) when divisor is positive integer and dividend is positive integer.
This topic is syllabus of cbse board class 10th mathematics. I have first given statement of Euclid’s division algorithm then taken few examples based on the statement. Euclid’s division lemma (algorithm) is very useful to solve problems based on Highest common factor or hcf or gcd (greatest common divisor) which is topic of real numbers (number theory). You can find it in ncert text book as theorem 1.1 i.e. What is the proof of Euclid’s division lemma along with proof and statement.
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[Divyansh-Tiwari]

Hi! as you showed here, while dividing -7 by 3 your quotient came out to be, -3 but -7/3 = can you explain that a little.

[OmShashwat]

Sir Euclids Division Lemma ke kuch questions kar ke example dijiye naa please