Why must q be the least element for (q+1)a to be greater than b?

In summary, the conversation discusses the existence part of a proof involving natural numbers. It is shown that there exists a unique pair (q,r) where q is the least element and r is in the range of 0 to a-1, such that b=aq+r. The conversation also discusses the relationship between q and b, and how q must be the least element for (q+1)a to be greater than b.
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Mathematicsresear
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Homework Statement


Let a, b be natural numbers then there exists a unique pair (q,r) that are elements of the non-negative integers such that b=aq+r and 0 is less than or equal to r which is less than a

I have a question regarding the existence part of the proof, now if I assumed a is less than b, its clear that there exists a positive integer x such that xa is greater than b. Now, why must q be the least element such that (q+1)a is greater than b?
 
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Assume the opposite and show that in that case r > a
 
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Mathematicsresear said:

Homework Statement


Let a, b be natural numbers then there exists a unique pair (q,r) that are elements of the non-negative integers such that b=aq+r and 0 is less than or equal to r which is less than a

I have a question regarding the existence part of the proof, now if I assumed a is less than b, its clear that there exists a positive integer x such that xa is greater than b. Now, why must q be the least element such that (q+1)a is greater than b?

If ##q## is the least element such that ##(q+1)a > b## then for all non-negative integers ##p \leq q## we have ##pa \leq b.## In particular, ##qa \leq b## but ##(q+1) a## is not ##\leq b##. That means that ##b-qa \in \{0,1,\ldots, a-1 \}.##
 

What is the Division Algorithm?

The Division Algorithm is a mathematical principle used to prove that any two integers, a and b, can be divided where the quotient and remainder are unique.

How is the Division Algorithm used to prove division?

The Division Algorithm uses the concept of division to show that any two integers, a and b, can be divided where the quotient and remainder are unique. It is a step-by-step process that shows the relationship between the dividend, divisor, quotient, and remainder.

What are the steps to prove division using the Division Algorithm?

The steps to prove division using the Division Algorithm are:

  1. Write the dividend and divisor in standard form (a = bq + r).
  2. Show that the remainder, r, is greater than or equal to 0 and less than the divisor, b.
  3. Divide the dividend by the divisor and find the quotient, q.
  4. Prove that the quotient and remainder are unique by showing that if the quotient and remainder are changed, the equation will no longer hold.

What are some common mistakes when using the Division Algorithm to prove division?

Some common mistakes when using the Division Algorithm include:

  • Not writing the dividend and divisor in standard form.
  • Forgetting to check that the remainder is greater than or equal to 0 and less than the divisor.
  • Incorrectly dividing the dividend by the divisor to find the quotient.
  • Not proving that the quotient and remainder are unique by showing that if they are changed, the equation will no longer hold.

How can the Division Algorithm be applied in real-life situations?

The Division Algorithm can be applied in real-life situations such as dividing a number of items equally between a certain number of people, finding the cost per unit when purchasing items in bulk, or calculating the number of days in a given number of weeks. It can also be used in more complex mathematical concepts, such as finding the greatest common divisor or simplifying fractions.

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