Question

दिखाइए कि किसी भी धन पूर्णांक का वर्ग 6m+2 या 6m+5 रे रूप का नहीं हो सकता, जहाँ m कोई पूर्णांक है
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Answer 1

Given :  धन पूर्णांक का वर्ग

Square of positive integer

To Find : दिखाइए कि  धन पूर्णांक का वर्ग 6m+2 या 6m+5 रे रूप का नहीं हो सकता

Square of any positive integer can not be of form 6m + 2or 6m + 5

Solution:

Without losing generality  any number can be Written in the form

6k , 6k+1 , 6k+2 , 6k+3 , 6k+4 , 6k + 5

(6k)² = 36k²  = 6* 6k²  = 6m

(6k + 1)² = 36k² + 12k + 1  =  6(6k² + 2k) + 1  = 6m + 1

(6k + 2)² = 36k² + 24k + 4  =  6(6k² +4k) +4  = 6m + 4

(6k + 3)² = 36k² + 36k + 9  =  6(6k² + 6k+1) + 3 = 6m + 3

(6k + 4)² = 36k² + 48k + 16  =  6(6k² +8k + 2) +4  = 6m + 4

(6k + 5)² = 36k² + 60k + 25 =  6(6k² +10k + 4) +1  = 6m + 1

Hence we can notice that square of any positive integer is of form

6m , 6m+ 1 , 6m+3 , 6m+4 only

Hence

Square of any positive integer can not be of form 6m + 2or 6m + 5

किसी भी धन पूर्णांक का वर्ग 6m+2 या 6m+5 रे रूप का नहीं हो सकता

Learn More:

prove that square of any positive integer is of the form 9k or 9k+1 ...

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Answer 2

Answer:

Given :  धन पूर्णांक का वर्ग

Square of positive integer

To Find : दिखाइए कि  धन पूर्णांक का वर्ग 6m+2 या 6m+5 रे रूप का नहीं हो सकता

Square of any positive integer can not be of form 6m + 2or 6m + 5

Solution:

Without losing generality  any number can be Written in the form

6k , 6k+1 , 6k+2 , 6k+3 , 6k+4 , 6k + 5

(6k)² = 36k²  = 6* 6k²  = 6m

(6k + 1)² = 36k² + 12k + 1  =  6(6k² + 2k) + 1  = 6m + 1

(6k + 2)² = 36k² + 24k + 4  =  6(6k² +4k) +4  = 6m + 4

(6k + 3)² = 36k² + 36k + 9  =  6(6k² + 6k+1) + 3 = 6m + 3

(6k + 4)² = 36k² + 48k + 16  =  6(6k² +8k + 2) +4  = 6m + 4

(6k + 5)² = 36k² + 60k + 25 =  6(6k² +10k + 4) +1  = 6m + 1

Hence we can notice that square of any positive integer is of form

6m , 6m+ 1 , 6m+3 , 6m+4 only

Hence

Square of any positive integer can not be of form 6m + 2or 6m + 5

किसी भी धन पूर्णांक का वर्ग 6m+2 या 6m+5 रे रूप का नहीं हो सकता

Learn More:

prove that square of any positive integer is of the form 9k or 9k+1 ...

brainly.in/question/20017920

Prove that square of any positive even integer is always even ...

brainly.in/question/7353545