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P Pankaj Sanodiya
by euclid division lemma,we know that If a and b are two positive integers, then,by euclid division lemma  a = bq + r, 0  r  b Let b = 3 Therefore, r = 0, 1, 2 Therefore, a = 3q or a = 3q + 1 or a = 3q + 2 If a = 3q:  If a = 3q + 1 : If a = 3q + 2 : Therefore, the square of any positive integer is either of the form 3m or 3m + 1.

P Pankaj Sanodiya
Let a be any positive integer a=6q+r where 0 < or = r < 6 put r=1: a=6q+1 which is an odd integer r=3: a=6q+3 which is an odd integer r=5: a=6q+5 which is an odd integer therefore, any positive odd integer is of form 6q+1,6q+3 or 6q+5, where q is some integer.
The denominator is of the form 2a x 5b where a = 0 and b = 5. Therefore the given rational number will have a terminating decimal expansion.
As the decimal part of the given number is non-terminating and repeating, the number is rational but its denominator will have factors other than 2 and 5.
Since the decimal part of the given number is non-terminating and non-repeating we can conclude that the given number is irrational and cannot be written in the form  where p and q are integers.
The denominator is of the form 2a x 5b where a = 9 and b = 9. Therefore the given number is rational and has a terminating decimal expansion.
decimal expansions of  rational numbers are (i)  (ii)  (iv)  (vI)  (viii)  (ix)
The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.
The denominator is of the form 2a x 5b where a = 1 and b = 1. Therefore the given rational number will have a terminating decimal expansion.
The denominator is of the form 2a x 5b where a = 0 and b = 1. Therefore the given rational number will have a terminating decimal expansion.
The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.
The denominator is of the form 2a x 5b where a = 3 and b = 2. Therefore the given rational number will have a terminating decimal expansion.
The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.
The denominator is of the form 2a x 5b where a = 6 and b = 1. Therefore the given rational number will have a terminating decimal expansion.
The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.
The denominator is of the form 2a x 5b where a = 3 and b = 0. Therefore the given rational number will have a terminating decimal expansion.
Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.
Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.
Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. Since p and q are co-prime integers  will be rational, this contradicts the fact that   is irrational. This contradiction arises because our initial assumption that   is rational was wrong. Therefore  is irrational.
Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.
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