MHB Find the Smallest Integer Challenge

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The challenge is to find the smallest integer that is a perfect square and begins with the digits 3005. Calculations indicate that for even digits, the square root of 3005 leads to an integer of approximately 54.8177, while for odd digits, the square root of 30050 gives around 173.34. The smallest integer derived from these calculations is 5482, as its square equals 30052324. The discussion emphasizes that calculators can be used, but computer-generated solutions are not acceptable. The final result confirms that 5482^2 meets the criteria of starting with 3005.
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Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)
 
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anemone said:
Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)

we can have even number of digits.odd

if even then we have

sqrt 3005 * 10^2n = 54.8177 * 10^n
sqrt 3006* 10^2n = 54.8270 * 10^n

if odd digits then

sqrt 30050 * 10^n = 173.34 * 10^n

sqrt 30060 * 10^n= 173.37 * 10 ^n

from the 1st set we get sqrt 5482 as smallest where a digit is different and from the second at least

17335

so it is 5482^2 = 30052324
 
Hi kaliprasad, thanks for participating and your answer is correct!:)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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