2 條奧數

1. Given that the decinal part of 5+ √11 is A and the decimal part of 5 - √11 is B. Find A+B.
Let
5+ √11 =x+A
5-√11=y+B
where x=8,y=1
So 5+ √11 +5- √11=(x+y)+(A+B)
10= (x+y)+(A+B)
A+B=10-(x+y)=10-9=1
2
Let R be the remainder of 1x3x5x7x9x11x13x15x1 7x19 divided by 4. Find the value of R.
The remainder of 1x3 divided by 4 is 3
The remainder of 5x7 divided by 4 is 3
...
The remainder of 17x19 divided by 4 is 3
So
The remainder of 1x3x5x7x9x11x13x15x1 7x19 divided by 4
=The remainder of (3^5/4)
=3
R=3

2008-01-26 19:19:43 補充：
第二條用同餘概念去諗

1. Given that the decinal part of 5+ √11 is A and the decimal part of 5 - √11 is B. Find A+B.

Since 8 < 5 + √11 < 9
A = 5 + √11 - 8

Since 1 < 5 - √11 < 2
B = 5 - √11 - 1

From the above
A + B = (5 + √11 - 8) + (5 - √11 - 1)
A + B = (√11 - 3) + (4 - √11)
A + B = 1

2. Let R be the remainder of 1x3x5x7x9x11x13x15x17x19 divided by 4. Find the value of R.

The sum of 1x3x5x7x9x11x13x15x17x19 = 654,729,075
Since the last two digit is 75
R = 3