SOLUTION: What is the age of this monuments if the age of monument is over 50 years old ? it's exact age is the square number of 2 years ago and will be a cube number of 2 years hence.

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Question 935243: What is the age of this monuments if the age of monument is over 50 years old ? it's exact age is the square number of 2 years ago and will be a cube number of 2 years hence.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The age of the monument is a number greater than 50.
very old monuments, like the pyramids and Stonehenge are believed to be less than 5300 years old.
In 2 years the age of the monument will be a perfect cube greater than 50%2B2=52,
and 4 less than that number (the age 2 years ago) is a perfect square.
We can calculate cubes that are greater than 52, but less than 5500,
and could be the age of the monument 2 years from now.
(It is easy with a calculator, and not that bad without one).
Then, we can check if 4 less than each cube is a perfect square.
(I know that numbers that are perfect squares must end in 0, 1, 4, 5, 6, or 9,
and that will make it easy to detect numbers that are not squares,
even without a calculator. For those not eliminated, I can try factoring).

Numbers over 50 that are perfect cubes are:
4%5E3=64 --> 64-4=60 is not a square.
5%5E3=125 --> 125-4=121=11%5E2 is a perfect square.
If the monument will be 125 years old in 2 years,
then it is 125-2=highlight%28123%29 years old right now.

I checked to see if there were any other plausible solutions. There were none.
The next 13 cubes are listed below, and each one is eliminated easily enough, even without using a calculator.
6%5E3=216 --> 216-4=212 is not a square (wrong end digit).
7%5E3=343 --> 343-4=339=3%2A126=3%5E2%2A42 is not a square, because 42 is not a square.
8%5E3=512 --> 512-4=508 is not a square (wrong end digit).
9%5E3=729 --> 729-4=725=25%2A29=5%5E2%2A29 is not a square, because 29 is not a square.
10%5E3=1000 --> 1000-4=996=3%2A332 is not a square, because 3 is a factor, but 3%5E2 is not.
11%5E3=1331 --> 1331-4=1327 is not a square (wrong end digit).
12%5E3=1728 --> 1728-4=1724 is not a square, because it is greater than 40%5E2=1600 , and less than the closest square ending in 4 ,42%5E2=40%5E2%2B2%2A40%2A2%2B2%5E2=1600%2B160%2B4=1764 .
13%5E3=2197 --> 2197-4=2193 is not a square (wrong end digit).
14%5E3=2744 --> 2744-4=2740=10%2A274 is not a square, because 10 is a factor, but 10%5E2-100 is not.
15%5E3=3375 --> 3375-4=3371 is not a square. It is between 50%5E2=2500 and 60%5E2=3600 , and ends in 1 , so it could only be 51%5E2 or 59%5E2. It is neither, because 59%5E2=%2830-1%29%5E2=60%5E2-2%2A60%2A1%2B1%5E2=3600-120%2B1=3481 , and 51%5E2=%283%2A17%29%5E2=3%5E3%2A17%5E2 would be a multiple of 3 , while 3371 is not a multiple of 3 .
16%5E3=4096 --> 4096-4=4092 is not a square (wrong end digit).
17%5E3=4913 --> 4913-4=4909 is not a square, because it is greater than 70%5E2=4900 , but it is less than 71%5E2 .
18%5E3=5832 --> 5832-4=5828 is not a square (wrong end digit, and too large a number).