主题：【Help】A Math contest question -- steve

My nephew got stuck to a Math contest question. I would highly appreciate it if anyone could give a lift.

How many four-digit positive integers x are there with the property that x and 3x have only even digits? (One such number is x = 8002, since 3x = 24006 and each of x and 3x has only even digits.)

(A) 82 (B) 84 (C) 86 (D) 88 (E) 90

Thanks for all your help!

Each of 2, 4, 0, 0, and 6 is an even number.

I thought it was EVEN NUMBER OF DIGITS. I was wrong.

2*3*3*3 + 2*3 + 2*3 + 2*3

the last 2*3 is refer to combination of 68's

holp this help.

2*3*3*3 + 3*3 + 2*3 + 2*3 + 1 + 2 + 3 + 1 = 82

1st term for yxxx where x is either 0,2,or 8 and y is either 2 or 8.

2nd for 68xx

3rd for y68x

4th for yx68

5th for 6868

6th for y668

7th for 668x

8th for 6668.

I can only think of the 2*3*3*3=54 integers made of 0,2,8;

What are the other 36 integers?

hua

2000 8000 2200 8200 2800 6800 8800 2020 8020 2220 8220 2820 6820 8820 2080 8080 2280 8280 2680 6680 8680 2880 6880 8880 2002 8002 2202 8202 2802 6802 8802 2022 8022 2222 8222 2822 6822 8822 2082 8082 2282 8282 2682 6682 8682 2882 6882 8882 2008 8008 2208 8208 2808 6808 8808 2028 8028 2228 8228 2828 6828 8828 2068 8068 2268 8268 2668 6668 8668 2868 6868 8868 2088 8088 2288 8288 2688 6688 8688 2888 6888 8888

complicated. so i still think 90 is the right answer. for example 88*3 = 264.

sorry

只需考虑一点，仅有0，2，8乘3，或者6乘3并且后一位进位是2（68，668，6668）时，进位是偶数（0或2）。随园的列举应该没问题。