Question 11
Write whether every positive integer can be of the form 4q+2 where q is an integer, Justify your answer
Solution:
To determine whether every positive integer can be expressed in the form 4q+2 where q is an integer, we will analyze the possible forms of integers based on the division by 4.
1. Understanding the Division Algorithm:
According to Euclid’s division algorithm, any integer a can be expressed as:
a=4m+r
where m is an integer and r is the remainder when a is divided by 4. The possible values for r are 0,1,2, or 3.
2. Identifying Forms of Integers:
Based on the value of r, we can categorize integers as follows:
– If r=0: a=4m (multiple of 4)
– If r=1: a=4m+1 (one more than a multiple of 4)
– If r=2: a=4m+2 (two more than a multiple of 4)
– If r=3: a=4m+3 (three more than a multiple of 4)
3. Analyzing the Form 4q+2:
The form 4q+2 corresponds to integers where the remainder r=2. This means that there are indeed integers that can be expressed in this form, specifically those integers that are two more than a multiple of 4.
4. Conclusion:
However, not every positive integer can be expressed in the form 4q+2. For example:
– The integer 1 can be expressed as 4(0)+1.
– The integer 3 can be expressed as 4(0)+3.
– The integer 4 can be expressed as 4(1)+0.
– The integer 5 can be expressed as 4(1)+1.
– The integer 6 can be expressed as 4(1)+2.
– The integer 7 can be expressed as 4(1)+3.
– The integer 8 can be expressed as 4(2)+0.
– The integer 9 can be expressed as 4(2)+1.
– The integer 10 can be expressed as 4(2)+2.
– The integer 11 can be expressed as 4(2)+3.
– The integer 12 can be expressed as 4(3)+0.
From this analysis, we see that integers can take on various forms, and only those integers that give a remainder of 2 when divided by 4 can be expressed as 4q+2. Therefore, not every positive integer can be of the form 4q+2.
Final Answer:
No, not every positive integer can be expressed in the form 4q+2, as only those integers that give a remainder of 2 when divided by 4 can be expressed in this form.