Let 
be any positive integer
We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying,
where
.
Take 
Since 0 ≤ <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">r </i>< 4, the possible remainders are 0, 1, 2 and 3.
That is, 
<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>can be 
, where <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>is the quotient.
Since <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>is odd, <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>cannot be 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>or 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 2 as they are both divisible by 2.
Therefore, any odd integer is of the form 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 1 or 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 3.
Test
Gaurav Seth 6 years, 7 months ago
Let
be any positive integer
We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying,
where
.
Take
Since 0 ≤ <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">r </i>< 4, the possible remainders are 0, 1, 2 and 3.
That is,
<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>can be 
, where <i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>is the quotient.
Since
<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>is odd, 
<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden"> </i>cannot be 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>or 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 2 as they are both divisible by 2.
Therefore, any odd integer is of the form 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 1 or 4<i style="box-sizing:inherit; outline:none; user-select:initial !important; line-height:inherit; max-width:100%; overflow:hidden">q </i>+ 3.
0Thank You