The teacher explains, "Now take out your homework class, and answer these ten 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
Alex panics, because he was too busy last night solving other things on the internet and didn't get a chance to do his homework. But when he looks over the quiz (where
exactly one answer is correct per question), he observes 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.
Solution to the Problem:
Alex figures that the number must be 357, and the 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
Explanation:
Looking at question 1, Alex deduces the number must be three digits between 100 and 899, otherwise A and B would be true.
Looking at question 3 and question 5, Alex eliminates divisibility by 15 because that would mean the number was divisible by both 3 and 5.
Looking at question 3 and question 6, Alex sees that the answer to 6 must be A. If it were divisible by 20, then the answer to 3 would be A and C.
Looking at question 4 and question 6, Alex notes the possible combinations for the number are:
_19
_38
_57
_76
_95
He eliminates _19 and _95 because there is no possible first digit.
So the possible numbers are:
138, 238, 338, 438, 538, 638, 738, 838, 938, or
157, 257, 357, 457, 557, 657, 757, 857, 957, or
176, 276, 376, 476, 576, 676, 776, 876, 976.
From question 4, Alex knows that the digits must be either increasing or decreasing.
So, that leaves only the following possible numbers:
138, 238, 157, 257, 357, 457, 876, and 976.
Now check divisibilty by 2, 3, and 5 (Question 3):
138 -- divisible by 2 and 3
238 -- divisible by 2
157 -- not divisible by 2, 3, or 5
257 -- not divisible by 2, 3, or 5
357 -- divisible by 3
457 -- not divisible by 2, 3, or 5
876 -- divisible by 2 and 3
976 -- divisible by 2
So, this narrows it to three possibilities: 238, 357, and 976.
Now looking at question 5:
238 -- divisible by 17
357 -- divisible by 17
976 -- not divisible by 15, 17, or 19
So, that eliminates 976 and leaves only 238 and 357 as possibilities.
Alex sees that two questions disallow 238 from being the answer. Question 7 and question 9 both have answers that aren't listed.
Two hundred thirty eight has no J, L, Q, V, X, or Z, and only has two occurrences of the letter E.
The number THREE HUNDRED FIFTY SEVEN contains the letter V (Question 7) and it contains 5 occurrences of the letter E (question 9).
Since 357 works with all the questions, it is the number Alex picked.
Correctly solved by:
| 1. Davit Banana | Istanbul, Turkey |
| 2. Sudhir Bavdekar | Mumbai, India |
| 3. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 4. Chris Walton |
Stone Bridge School, Chesapeake, Virginia |
| 5. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 6. Seth Cohen | Concord, New Hampshire, USA |
| 7. Alex |
Muckleshoot Tribal School, Auburn, Washington, USA |
| 8. Eddie |
Muckleshoot Tribal School, Auburn, Washington, USA |