document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #568422 by KMST(5328) $\"\"$ $\"About$

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The age of the monument is a number greater than 50.
\n" ); document.write( "very old monuments, like the pyramids and Stonehenge are believed to be less than 5300 years old.
\n" ); document.write( "In 2 years the age of the monument will be a perfect cube greater than $\"50%2B2=52\"$ ,
\n" ); document.write( "and 4 less than that number (the age 2 years ago) is a perfect square.
\n" ); document.write( "We can calculate cubes that are greater than 52, but less than 5500,
\n" ); document.write( "and could be the age of the monument 2 years from now.
\n" ); document.write( "(It is easy with a calculator, and not that bad without one).
\n" ); document.write( "Then, we can check if 4 less than each cube is a perfect square.
\n" ); document.write( "(I know that numbers that are perfect squares must end in 0, 1, 4, 5, 6, or 9,
\n" ); document.write( "and that will make it easy to detect numbers that are not squares,
\n" ); document.write( "even without a calculator. For those not eliminated, I can try factoring).
\n" ); document.write( "
\n" ); document.write( "Numbers over 50 that are perfect cubes are:
\n" ); document.write( " $\"4%5E3=64\"$ --> $\"64-4=60\"$ is not a square.
\n" ); document.write( " $\"5%5E3=125\"$ --> $\"125-4=121=11%5E2\"$ is a perfect square.
\n" ); document.write( "If the monument will be 125 years old in 2 years,
\n" ); document.write( "then it is $\"125-2=highlight%28123%29\"$ years old right now.
\n" ); document.write( "
\n" ); document.write( "I checked to see if there were any other plausible solutions. There were none.
\n" ); document.write( "The next 13 cubes are listed below, and each one is eliminated easily enough, even without using a calculator.
\n" ); document.write( " $\"6%5E3=216\"$ --> $\"216-4=212\"$ is not a square (wrong end digit).
\n" ); document.write( " $\"7%5E3=343\"$ --> $\"343-4=339=3%2A126=3%5E2%2A42\"$ is not a square, because $\"42\"$ is not a square.
\n" ); document.write( " $\"8%5E3=512\"$ --> $\"512-4=508\"$ is not a square (wrong end digit).
\n" ); document.write( " $\"9%5E3=729\"$ --> $\"729-4=725=25%2A29=5%5E2%2A29\"$ is not a square, because $\"29\"$ is not a square.
\n" ); document.write( " $\"10%5E3=1000\"$ --> $\"1000-4=996=3%2A332\"$ is not a square, because $\"3\"$ is a factor, but $\"3%5E2\"$ is not.
\n" ); document.write( " $\"11%5E3=1331\"$ --> $\"1331-4=1327\"$ is not a square (wrong end digit).
\n" ); document.write( " $\"12%5E3=1728\"$ --> $\"1728-4=1724\"$ is not a square, because it is greater than $\"40%5E2=1600\"$ , and less than the closest square ending in $\"4\"$ , $\"42%5E2=40%5E2%2B2%2A40%2A2%2B2%5E2=1600%2B160%2B4=1764\"$ .
\n" ); document.write( " $\"13%5E3=2197\"$ --> $\"2197-4=2193\"$ is not a square (wrong end digit).
\n" ); document.write( " $\"14%5E3=2744\"$ --> $\"2744-4=2740=10%2A274\"$ is not a square, because $\"10\"$ is a factor, but $\"10%5E2-100\"$ is not.
\n" ); document.write( " $\"15%5E3=3375\"$ --> $\"3375-4=3371\"$ 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 $\"3\"$ .
\n" ); document.write( " $\"16%5E3=4096\"$ --> $\"4096-4=4092\"$ is not a square (wrong end digit).
\n" ); document.write( " $\"17%5E3=4913\"$ --> $\"4913-4=4909\"$ is not a square, because it is greater than $\"70%5E2=4900\"$ , but it is less than $\"71%5E2\"$ .
\n" ); document.write( " $\"18%5E3=5832\"$ --> $\"5832-4=5828\"$ is not a square (wrong end digit, and too large a number). \n" ); document.write( "

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