February 2016
Problem of the Month

Three Digit Number problem
by Paddy Smith in GAMES

A certain three-digit number is the product of two different
prime numbers that end with the same digit.

The three digit number's middle digit is larger than its first
digit and smaller than its third digit.
(In other words, the tens digit is larger than its hundreds digit
and smaller than its ones digit).

What is the three-digit number?

Solution to the Problem:

There are nine answers:
129 = 3 * 43
159 = 3 * 53
249 = 3 * 83
259 = 7 * 37
469 = 7 * 67
489 = 3 * 163
679 = 7 * 97
689 = 13 * 53
789 = 3 * 263

Marco Morelli sent in this analysis:
First of all, all prime numbers (except 2 and 5) end with 1 or 3 or 7 or 9;

however, the last digit of the two prime numbers cannot be 1 nor 9, otherwise their product would have 1 as the units digit, and a three-digit number having its hundreds digit smaller than its tens digit smaller than 1 cannot exist;

so we are left with 3 or 7;

now we must assume that the two prime numbers have at least 2 digits, so we end up with evaluating the following couples:

13*23 = 299
13*43 = 559
13*53 = 689
13*73 = 949

17*37 = 629
17*47 = 799

which are the only couples having a three-digit number as their product;

the desired number is therefore 689.

Notice that if we drop that assumption, there are other couples, such as:

3*43 = 129
3*53 = 159
3*163 = 489
3*193 = 579
3*263 = 789

7*37 = 259
7*67 = 469
7*97 = 679

Correctly solved by:

1. James Alarie (sent in all 9 solutions)	Flint, Michigan
2. Chris White	St. Marks Elementary, Colwich, Kansas
3. Celton Perry (sent in 2 solutions)	Mountain View High School, Mountain View, Wyoming
4. SreeRoopa Sankararaman	Singapore, Singapore
5. Marco Morelli (sent in all 9 solutions)	Fermo, Italy
6. Chris Striniger	Mountain View High School, Mountain View, Wyoming
7. Keagan Genzer	Mountain View High School, Mountain View, Wyoming
8. Breanna Hood	Mountain View High School, Mountain View, Wyoming
9. Charity Harmon	Mountain View High School, Mountain View, Wyoming
9. Kalen Rhodes	St. Joseph High School, Westchester, Illinois
10. Chelsey M.	John F Kennedy High School, Taylor, Michigan
11. Tristan Hamblin	Mountain View High School, Mountain View, Wyoming
12. Alyx Wells	Mountain View High School, Mountain View, Wyoming
13. Jessica Hamblin	Mountain View High School, Mountain View, Wyoming
14. Trenton Harmon	Mountain View High School, Mountain View, Wyoming
15. Jessica Hamblin	Mountain View High School, Mountain View, Wyoming
16. Mckinzee Walker	Mountain View High School, Mountain View, Wyoming
17. Samantha Brailsford	Mountain View High School, Mountain View, Wyoming
18. Alexis Stoddard	Mountain View High School, Mountain View, Wyoming
19. Hannah Stoddard	Mountain View High School, Mountain View, Wyoming
20. Cheyenne Dunn	Mountain View High School, Mountain View, Wyoming
21. Gage Barney	Mountain View High School, Mountain View, Wyoming
22. Dusty Q. Iorg	Mountain View High School, Mountain View, Wyoming
23. Nasima Madbouhi	Mountain View High School, Mountain View, Wyoming
24. Kayle Rippetoe	Mountain View High School, Mountain View, Wyoming
25. Mashayla Hurdsman	Mountain View High School, Mountain View, Wyoming
26. Kaylee Gross	Mountain View High School, Mountain View, Wyoming
27. Taylor Meeks	Mountain View High School, Mountain View, Wyoming

Send any comments or questions to: David Pleacher