I am a three digit number.
I am either divisible by 3 or by 5.
I am either divisible by 4 or by 6.
I am either divisible by 5 or by 7.
I am either divisible by 6 or by 8.
I am either divisible by 7 or by 9.
I am either divisible by 9 or by 11.
What am I?
(Note: "divisible" means leaving no remainder.)
Solution to the Problem:
The three-digit number is 462.Assume that the number is divisible by 9. This would mean (from the clues above):
It is NOT divisible by 11.
It is NOT divisible by 7.
It IS divisible by 5.
It is NOT divisible by 3.
However this is a contradiction to the assumption. If a number is divisible by 9 (our assumption), then it must be divisible by 3, but we just showed that it is NOT divisible by 3, so the assumption is false.
Therefore, the number is NOT divisible by 9, so
it is divisible by 11,
it is divisible by 7,
so it is NOT divisible by 5,
and if it is not divisible by 5, it must be divisible by 3.
(all these statements came from the clues above)
Therefore, the number is divisible by 3, 7, and 11, but not by 5 or 9.
The three-digit numbers that are multiples of 3, 7, and 11 are 231, 462, 693, and 924.
693 is divisible by 9, so that cannot be correct.
The correct answer must be divisible by either 4 or 6 (clue #3);
231 is divisible by neither, and 924 is divisible by both.
Therefore the only number that works is 462 (it is divisible by 6, but not by 4 or 8).
Correctly solved by:
| 1. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 2. Davit Banana | Istanbul, Turkey |
| 3. Kamal Lohia |
Holy Angel School, Hisar, Haryana, India |
| 4. Ivy Joseph | Pune, Maharashtra, India |
| 5. Sudhir Bavdekar | Mumbai, India |
| 6. Seth Cohen | Concord, New Hampshire, USA |
| 7. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 8. Kelly Stubblefield | Mobile, Alabama, USA |