Page 43 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

1.114 x 1 ,x 2 ,x 3 ,x 4 ,x 5 are distinct positive odd numbers and satisfy
(2005 − x 1 )(2005 − x 2 )(2005 − x 3 )(2005 − x 4 )(2005 − x 5 ) = 576, what is the last digit of

x + x + x + x + x ?
2
2
2
2
2
4
5
2
1
3
Solution: Since x 1 ,x 2 ,x 3 ,x 4 ,x 5 are distinct positive odd numbers, then
2005 − x 1 , 2005 − x 2 , 2005 − x 3 , 2005 − x 4 , 2005 − x 5 are distinct even numbers, thus
576 needs to be factored into the product of five distinct even numbers, which has a
unique form: 576 = 24 =2 × (−2) × 4 × 6 × (−6) . Hence,
2
2
2
2
2
2
2
2
(2005−x 1 ) +(2005−x 2 ) +(2005−x 3 ) +(2005−x 4 ) +(2005−x 5 ) =2 +(−2) +
2
2
2
2
2
2
2
2
2
4 +6 +(−6) = 96 ⇒ 5×2005 −4010(x 1 +x 2 +x 3 +x 4 +x 5 )+(x +x +x +x +x ) =
1
5
2
3
4
2
2
2
2
2
2
96 ⇒ x + x + x + x + x = 96 − 5 × 2005 + 4010(x 1 + x 2 + x 3 + x 4 + x 5 ) ≡ 1
5
1
2
4
3
(mod 10), that is, the last digit of x + x + x + x + x is 1.
2
2
2
2
2
5
3
1
2
4
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