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Common Factors

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prove that If a,b,c are natural numbers such that c divides ab and c & a are co-prime then c divides b.

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Q. Call a natural number

n

faithful, if there exist numbers

a < b < c

a

b, b

c

n = a + b + c

. Show that all but a finite number of natural numbers are faithful. Find the sum of all natural numbers which are not faithful.

Q. If a + b + c +ab +bc + ca + abc = 2005 where a, b, c are distinct natural number such that a < b < c then

a

b

are natural numbers such that

a < b

\frac{a}{b}

is a/an fraction.

a

b

are natural numbers such that

a^{2} - b^{2}

is prime number, then

a^{2} + b^{2} - 1

If a, b, c are rational numbers such that a² + b² + c² − ab − bc − ca = 0, prove that a = b = c.

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