How to Solve Euclid Division Questions

Euclid's Division Lemma is a fundamental concept in number theory that states that for any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, where 0 ≤ r < />

Euclid's Division Lemma is often used to solve division questions in number theory. To solve Euclid division questions, follow the steps below:

Step 1: Understand the Problem
- Read the question carefully and identify the given values and what needs to be solved.
- Determine the values of a and b in the equation a = bq + r.

Step 2: Apply Euclid's Division Lemma
- Apply Euclid's Division Lemma to find the quotient (q) and remainder (r) when a is divided by b.
- Use the formula a = bq + r and substitute the given values of a and b to solve for q and r.
- Ensure that the remainder (r) satisfies the condition 0 ≤ r < />

Step 3: Verify the Solution
- Check if the obtained values of q and r satisfy the equation a = bq + r.
- Substitute the values of q and r into the equation and verify if it holds true.

Example:
Let's solve an example to illustrate the steps mentioned above.

Question: Find the quotient and remainder when 35 is divided by 7.

Step 1: Understand the Problem
- The given values are a = 35 and b = 7.

Step 2: Apply Euclid's Division Lemma
- Using the formula a = bq + r, substitute the given values:
35 = 7q + r

- To find the quotient and remainder, we need to determine the values of q and r.

Step 3: Verify the Solution
- We obtained q = 5 and r = 0.
- Substituting these values into the equation: 35 = 7(5) + 0, we can verify that it holds true.

Therefore, the quotient is 5 and the remainder is 0 when 35 is divided by 7.

By following these steps and applying Euclid's Division Lemma, you can solve Euclid division questions efficiently.

With your hands and with pencil/pen on paper/book