Any ____________ is of the form 4q + 1 or 4q + 3 for some integer ‘q’.a)composite numberb)positive odd integerc)prime numberd)positive even integerCorrect answer is option 'B'. Can you explain this answer?

Question

Any ____________ is of the form 4q + 1 or 4q + 3 for some integer &#8216;q&#8217;.a)composite numberb)positive odd integerc)prime numberd)positive even integerCorrect answer is option 'B'. Can you explain this answer?

Answer

Are u a CBSE topper ?

Answer

Hìi Classmate here is ur answer.          4q +1 & 4q+3 is  positive odd integer bcoz it cannot be divisible by 2.THATS SOLVEUPVOTE AND FOLLOW ME FOR ANY HELP

Answer

It can be proved. But it is a sort of long process

Answer

Explanation:

Positive Odd Integer:
- An odd integer is of the form 2q + 1 for some integer q.
- If we substitute q = 2p, where p is an integer, we get 4p + 1, which is of the form 4q + 1.
- Similarly, if we substitute q = 2p + 1, we get 4p + 3, which is of the form 4q + 3.

Reasoning:
- Any positive odd integer can be expressed in the form 4q + 1 or 4q + 3.
- This is because odd integers are always one more or one less than multiples of 2, and multiples of 2 are of the form 4q.
- Therefore, any positive odd integer can be represented in the given forms.

Conclusion:
- The correct answer is option 'B': positive odd integer.
- This is because any positive odd integer fits the criteria of being of the form 4q + 1 or 4q + 3 for some integer q.

Answer

Let a be a given positive odd integer.
Applying Euclid’s Division Lemma to aa and b = 4,,
We have, a = 4q+r where 0 ⩽ r < 4 ⇒ r = 0, 1, 2, 3 ⇒ a = 4q or 4q+1 or 4q+2 or 4q+3
But a = 4q and 4q+2 = 2(2q+1)are clearly even.
Also a = 4q, 4q+1, 4q+2, 4q+3 are consecutive integers, therefore any positive odd integer is of the form 4q+1 and 4q+3 where q is some integer.

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