Page 10 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

1.14 If x, y are positive integers and satisfy 2 × x = 25xy where 25xy is one number
y
5
instead of a multiplication, find the values of x and y .

Solution: Since 2 is an even number, 25xy is even, thus y can only be 2, 4, 6, 8. When
5
y =2, we consider two cases: If x< 9, then 2 x ≤ 2 × 8 = 2048 not in the structure
5 2
2
5
of 25xy ; If x =9, we have 2 × 9 = 2592 within the structure of 25xy . Hence, x =9,y =2
5
2
satisfy all the conditions. Similarly we can discuss the cases y =4, 6, 8, and we find that
no x value satisfies all the conditions.
360°
2
2
1.15 The real numbers a, b, c satisfy a + b + c =9, what is the maximum of
2
2
2
(a − b) +(b − c) +(c − a) ? thinking.
2
360°
2
2
2
2
2
2
2
2
Solution: (a − b) +(b − c) +(c − a) = 2(a + b + c ) − (2ab +2bc +2ca)=3(a + b +
2
2
2
2
2
2
2
2
(a − b) +(b − c) +(c − a) = 2(a + b + c ) − (2ab +2bc +2ca)=3(a + b +
2
2
c ) − (a + b + c)
thinking.
(a − b) +(b − c) +(c − a) = 2(a + b + c ) − (2ab +2bc +2ca)=3(a + b +c ) − (a + b + c) . Since a, b, c are real numbers, (a + b + c) ≥ 0. In addition, a + b + c =9.
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c ) − (a + b + c) 2 Thus (a − b) +(b − c) +(c − a) ≤ 3(a + b + c ) =3 × 9 = 27. The maximal value is 27.
2
2
2
2
2
2
360°
thinking.
360°
thinking.
Discover the truth at www.deloitte.ca/careers Discover the truth at www.deloitte.ca/careers

© Deloitte & Touche LLP and affiliated entities.
Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities.

© Deloitte & Touche LLP and affiliated entities.
Discover the truth at www.deloitte.ca/careers
Download free eBooks at bookboon.com Click on the ad to read more
Click on the ad to read more
10
10
© Deloitte & Touche LLP and affiliated entities.