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3 AND 4.....
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Answer:
remaining questions of no. 3 will be solved like first no. of 3
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QUESTION 1) Find the least number which must be added to the following numbers to them perfect square.
1) 5607
⠀⠀⠀⠀⠀74⠀⠀⠀
⠀7 ⠀| 56 07
⠀⠀⠀| 49⠀⠀⠀⠀
144⠀|⠀⠀707
⠀⠀⠀|⠀⠀576⠀⠀
⠀⠀⠀|⠀⠀131
Hence, the Remainder = 131
Since, We know that (74)² < 5607
Therefore , Let's take, the next perfect square number which is (75)² . Now , (75)² = 5625 > 5607
Therefore, the number to be added
5625 - 5607 = 18
2) 4931
⠀⠀⠀⠀⠀70⠀⠀⠀
⠀7 ⠀| 49 31
⠀⠀⠀| 49⠀⠀⠀⠀
140⠀|⠀⠀31
⠀⠀⠀|⠀⠀⠀0⠀⠀
⠀⠀⠀|⠀⠀31
Hence, the Remainder = 31
Since, We know that (70)² < 4931
Therefore , Let's take, the next perfect square number which is (71)² . Now , (71)² = 5041 > 4931
Therefore, the number to be added
5041 - 4931 = 110
3) 4515600
⠀⠀⠀⠀⠀2124⠀⠀⠀⠀
⠀2 ⠀⠀| 45 15 600
⠀⠀⠀⠀| 4⠀⠀⠀⠀⠀⠀⠀
⠀41⠀⠀|⠀51
⠀⠀⠀⠀|⠀41⠀⠀⠀⠀⠀⠀
422 ⠀|⠀⠀1056
⠀⠀⠀⠀|⠀⠀⠀844⠀⠀⠀
4244⠀|⠀⠀⠀21200
⠀⠀⠀⠀|⠀⠀⠀16976⠀⠀
⠀⠀⠀⠀|⠀⠀⠀⠀4224
Hence, the Remainder = 4224
Since, We know that (2124)² < 4515600
Therefore , Let's take, the next perfect square number which is (2125)² . Now , (2125)² = 4515625 > 4515600
Therefore, the number to be added
4515625 - 4515600 = 25
4) 37460
⠀⠀⠀⠀⠀194⠀⠀⠀⠀
⠀1 ⠀⠀| 37 460
⠀⠀⠀⠀| 1⠀⠀⠀⠀⠀⠀
⠀29 ⠀|⠀274
⠀⠀⠀⠀|⠀261⠀⠀⠀⠀
384 ⠀|⠀⠀1360
⠀⠀⠀⠀|⠀⠀1536⠀⠀
⠀⠀⠀⠀|⠀⠀⠀- 176
Hence, the Remainder = 176
Since, We know that 37460 is 176 more than (194)²
Therefore, the number to be added will be 176
5) 506900
⠀⠀⠀⠀⠀711⠀⠀⠀
⠀7 ⠀⠀| 50 69 00
⠀⠀⠀⠀| 49⠀⠀⠀⠀
141⠀ ⠀|⠀169
⠀⠀⠀⠀|⠀141⠀⠀⠀⠀
1422 ⠀|⠀⠀2800
⠀⠀⠀⠀|⠀⠀2844⠀⠀
⠀⠀⠀⠀|⠀⠀⠀- 44
Hence, the Remainder = 44
Since, required no = 712
Therefore , Let's take, the next perfect square number which is (712)² .
Therefore, the number to be added
506944 - 506900 = 44
______________________________________________
QUESTION 2) find the greatest number of 5 digit which is a perfect square
the greatest 5 digit number is 99999
therefore ,
⠀⠀⠀⠀⠀2124⠀⠀
⠀3 ⠀| 99 99 9
⠀⠀⠀| 9⠀⠀⠀⠀⠀
⠀61⠀|⠀99
⠀⠀⠀|⠀61⠀⠀⠀⠀
626 |⠀⠀3899
⠀⠀⠀|⠀⠀3756⠀
⠀⠀⠀|⠀⠀⠀143
hence we must subtract 143 from 99999
Therefore , 99999 - 143 = 99856