The teacher explains, "Now take out your homework class, and answer these questions about the number you calculated for homework. Remember, it was a positive integer no more than 1000."

1 Is the number
A) Less than 100
B) Less than 900
C) Greater than 900

2 Is the sum of the digits
A) Greater than 10
B) Greater than 20
C) Greater than 30

3 Is the number divisible by:
A) 2
B) 3
C) 5

4 Are the digits:
A) Strictly increasing
B) Strictly decreasing

5 Is the number divisible by:
A) 15
B) 17
C) 19

6 If you get rid of the number's first digit, is it divisible by:
A) 19
B) 20

7 When spelled out, does the spelling use:
A) J
B) L
C) Q
D) V
E) X
F) Z

8 Ignoring the sign, what is the difference between the hundred's digit and the one's digit?
A) 2
B) 3
C) 4
D) 5

9) How many times does the letter E appear when the word is spelled out?
A) 3
B) 5
C) 7

10 Is the first digit of the number
A) Odd
B) Even

You panic, because you were too busy last night solving other things on flooble and didn't get a chance to do your homework. But when you figure out the quiz (where exactly one answer is right per question), you see that only one number will work.

What is this number?

(Note: Spelled out means in words. For example: 123 is "One hundred twenty three". Divisible means divided with no remainder. Also, padding the number with leading zeroes isn't allowed.)

357

First, we are told the the number in question is less than 1000. If it were less than 100, it would also be less than 900, and #1 would have two answers. Therefore, the answer to question one is B or C, and we have a three-digit number.

No three-digit number can have a sum of digits greater than 30 (or 27, for that matter). Also, if the sum of the number's digits were greater than 20, it would also be greater than 10, and #2 would have two answers. Therefore, the sum is greater than 10 and not greater than 20, or between 11 and 20 (inclusive), and the answer to #2 ia A.

If any number is divisible by 20, the ones place is a zero, and the tens place digit is odd. Thus, for number 6, if the last two digits were a mutliple of 20, the entire number would be divisible by 20, and 5 and 2 would both be answers to number 3. Therefore, the answer to question 6 is A.

Similarly, if the number were divisible by 15, it would be divisible by 3 and 5, and #3 would have two answers. Therefore, the answer to #5 is B or C.

Look at question 7. There are no numbers that contain J or Q, and zero is the only number that contains the letter Z.
Also, the only number less than one million that contains an L is eleven (or some number containing eleven), but then the number would also contain a V. Since there is exactly one correct answer to each question, that possiblity is out.
Therefore, the number contains a V (five or seven) or an X (six), and not both.
Note that while seventeen and seventy each contain a V, and sixteen and sixty each contain an X, fifteen and fifty do not contain a V, so numbers ending in 15, 56, or 57 are still valid.

We know that the last two digits of the number are a multiple of 19. There are five two-digit multiples of 19:
19
38
57
76
95

The digits are in strictly ascending or descending order (4), and the first and last digit do not differ by more than 5 (8).

Also recall that the number must have a 6 or 7 in any place, a 5 in the hundreds place, or a 5 in the ones place (unless preceded by a one, which we don't have to consider.

From these observations alone, and the listed possibilites for the last two digits, we can narrow it down to three numbers:
257
357
457

A moot observation at this point, we know that the answer to number one must be B, since these are all less than 900.

Each of these numbers has strictly increasing digits, so #4's correct answer is A.

Also, since each of the remaining numbers contains a 7 (V), and not a 6 (X), the answer to question seven must be D.

Finally, none these numbers is even or a multiple of five, so the answer must be a multiple of three, and the answer to question three is B.

Remembering the simple fact that the digits of any multiple of three add up to a multiple of three, we see that 357 (3+5+7=15) must be the number in question.

Going back, now, 357 is a multiple of 17, not 19, and the answer to #5 is B.
The difference between the first and last digit is, of course, 4, and the answer to #8 is C.
Also, E appears five times in 357 (thrEE hundrEd fifty sEvEn), so the answer to #9 is B.
Finally, the first digit (3) is odd.

The correct answers to the quiz are:

 1  B


 2  A


 3  B


 4  A


 5  B


 6  A


 7  D


 8  C


 9  B


10  A

Phew!
Edited on August 28, 2003, 10:27 pm

Posted by DJ on 2003-08-28 22:25:13