document.write( "Question 894462: 1.If the DOB of Grand son and Grand Father is same and for six consecutive years Grand Father's age in multiple of Grand Son's age what will be the age of Grand Son and Grand Father?\r
\n" ); document.write( "\n" ); document.write( "2.Prove that Root3+7 is irrational number \n" ); document.write( "

Algebra.Com's Answer #542176 by KMST(5328) $\"\"$ $\"About$

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1. A grandfather and grandson celebrate their birthdays on the same day of the year, and for 6 consecutive years the grandfather's age is a multiple of the grandson's age.
\n" ); document.write( "SOLUTION:
\n" ); document.write( "on the first of those 6 consecutive years the grandfather is 61, and the grandson is 1,
\n" ); document.write( "on the second year, their ages are 62 and 2;
\n" ); document.write( "on the third year, their ages are 63 and 3;
\n" ); document.write( "on the fourth year, their ages are 64 and 4;
\n" ); document.write( "on the fifth year, their ages are 65 and 5, and
\n" ); document.write( "on the sixth year, the grandfather is 66, and the grandson is 6.
\n" ); document.write( "REASONING:
\n" ); document.write( "Starting with a very young grandson makes sense, so we start with a 1 year old grandson, who will be 2, 3, 4, 5, and 6 in the next 5 consecutive years.
\n" ); document.write( "If the grandfather's age on the first year is $\"x\"$ ,
\n" ); document.write( "it is a multiple of the grandson's age, which is $\"1\"$ ,
\n" ); document.write( "but in the next 5 years it must be that
\n" ); document.write( " $\"x%2B1\"$ is a multiple of 2;
\n" ); document.write( " $\"x%2B2\"$ is a multiple of 3;
\n" ); document.write( " $\"x%2B3\"$ is a multiple of 4;
\n" ); document.write( " $\"x%2B4\"$ is a multiple of 5, and
\n" ); document.write( " $\"x%2B5\"$ is a multiple of 6.
\n" ); document.write( "That weans that $\"x-1\"$ is a multiple of 2, 3, 4, 5, and 6.
\n" ); document.write( "The least common multiple of those numbers is $\"60\"$ .
\n" ); document.write( " $\"x-1=60\"$ ---> $\"x=61\"$ gives the grandfather a reasonable age.
\n" ); document.write( "COMMENT:
\n" ); document.write( "Other common multiples of 2, 3, 4, 5, and 6, like 120, 180, and so on, make the grandfather way too old.
\n" ); document.write( "As another unrealistic option, we could start with a grandson who is 2 and a grandfather whose age minus 2 is a multiple of 2, 3, 4, 5, 6 and 7. Since the least common multiple is 210, that would make the grandfather at least 212. Making the grandson even older, makes the grandfather even more ridiculously old.
\n" ); document.write( "
\n" ); document.write( "2. $\"sqrt%283%29%2B7\"$ is irrational, because $\"sqrt%283%29\"$ is irrational.
\n" ); document.write( "If $\"sqrt%283%29%2B7\"$ were rational, $\"sqrt%283%29%2B7%2B%28-7%29=sqrt%283%29\"$ , being the sum of two rational numbers would be rational.
\n" ); document.write( "The fact that $\"sqrt%283%29\"$ is irrational is proven like the fact that $\"sqrt%282%29\"$ is irrational.
\n" ); document.write( "If $\"sqrt%283%29\"$ were rational, it could be written as an irreducible fraction,
\n" ); document.write( " $\"sqrt%283%29=m%2Fn\"$ with $\"m\"$ and $\"n\"$ not having any common factors.
\n" ); document.write( "Then,
\n" ); document.write( " $\"sqrt%283%29=m%2Fn\"$ ---> $\"%28sqrt%283%29%29%5E2=%28m%2Fn%29%5E2\"$ ---> $\"3=m%5E2%2Fn%5E2\"$ ---> $\"3n%5E2=m%5E2\"$ .
\n" ); document.write( "That means that $\"m%5E2\"$ is a multiple of 3,
\n" ); document.write( "but then $\"m\"$ must be a multiple of 3,
\n" ); document.write( "which would make $\"m%5E2\"$ a multiple of 9, and $\"m%5E2%2F3=n%5E2\"$ would be a multiple of 3, just like $\"m\"$ is.
\n" ); document.write( "That contradicts the starting hypothesis that $\"sqrt%283%29=m%2Fn\"$ with $\"m\"$ and $\"n\"$ not having any common factors.
\n" ); document.write( "So, it is impossible to find a rational number $\"m%2Fn\"$ with $\"m\"$ and $\"n\"$ not having any common factors, and $\"%28m%2Fn%29%5E2=3\"$ .
\n" ); document.write( "In other words, it is impossible to find a rational number equal to $\"sqrt%283%29\"$ , so $\"sqrt%283%29\"$ is irrational. \n" ); document.write( "

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