Page 40 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

2
2
2
2
2
2
2
2
2 2
2
2
2 2
(1 − b )(1 − c ) (1 − a )(1 − c ) (1 − a )(1 − b ) 1 − b − c + b c 1 − a − c + a c
+ + = + +
bc ac ab bc ac
2 2
2
2
1 − a − b + a b 1 1 1 b + c a + c a + b a + b + c
=( + + )− − − +ab+ac+bc = −
ab bc ac ab a b c abc
(bc−1)−(ac−1)−(ab−1)+ab+ac+bc =1−bc+1−ac+1−ab+1+ab+ac+bc =4 .
1.107 Let a, b, c be distinct positive integers, show at least one of
3
3
a b − ab ,b c − bc ,c a − ca is divisible by 10.
3
3
3
3
3
2
2
Proof: Because a b − ab = ab(a − b ),b c − bc = bc(b − c ),c a − ca = ca(c − a ) ,
3
2
3
2
3
3
3
2
2
then if a, b, c has at least one even number or they are all odd numbers, a b − ab ,b c − bc ,c a − ca 3
3
3
3
3
3
are divisible by 2.
If one of a, b, c is a multiple of 5, then the conclusion is proven.

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