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Any ____________ is of the form 4q + 1 or 4q + 3 for some integer ‘q’.
  • a)
    composite number
  • b)
    positive odd integer
  • c)
    prime number
  • d)
    positive even integer
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Any ____________ is of the form 4q + 1 or 4q + 3 for some integer R...
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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Most Upvoted Answer
Any ____________ is of the form 4q + 1 or 4q + 3 for some integer R...
Let a be any positive integer 
by EDL  a = bq +r 
0 ≤ r < b
possible remainders are 0, 1, 2 , 3 
 this shows that a can be in the form of 4q, 4q+1, 4q+2, 4q+3 q is quotient 
as a is odd a  can't be the form of 4q or 4q+2 as they are even
so a ill be in the form of 4q + 1 or 4q+3 
hence proved

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Community Answer
Any ____________ is of the form 4q + 1 or 4q + 3 for some integer R...
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
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