Divisibility puzzle : a*b*c | (a + b + c)^n, n = 7,13,21,31

 
Definition :

If   a   and   b   are integers (with   a   not zero),   we say   a divides   b

if there is an integer   c   such that   b \; = \; a \, c.

we write,   a | b   meaning   a   divides   b.

 
 

(1)   Let   a, \; b, \; c   be positive integers such that

a \; | \; b^2,
b \; | \; c^2 ,
c \; | \; a^2

Prove that   a \,b \,c \; | \; (a + b + c)^7

 
 

(2)   Let   a, \; b, \; c   be positive integers such that

a \; | \; b^3,
b \; | \; c^3,
c \; | \; a^3

Prove that   a \,b \,c \; | \; (a + b + c)^{13}

 
 

(3)   Let   a, \; b, \; c   be positive integers such that

a \; | \; b^4
b \; | \; c^4
c \; | \; a^4

Prove that   a \,b \,c \; | \; (a + b + c)^{21}

 
 

(4)   Let   a, \; b, \; c   be positive integers such that

a \; | \; b^5,
b \; | \; c^5,
c \; | \; a^5

Prove that   a \,b \,c \; | \; (a + b + c)^{31}

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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