T O P I C R E V I E W 
TchrWill 
Posted  07/08/2013 : 15:44:09 The digits of a 5 digit number ABCDE have the following relationships:
1The first two digits are the square of the third digit. 2The fourth and fifth digits are the sum of the second and third digits. 3The third digit is twice the second digit. 4The fifth digit is one half the second digit and one third the first digit. 5The sum of the five digits is 21. 6What is the 5 digit number?

2 L A T E S T R E P L I E S (Newest First) 
TchrWill 
Posted  07/29/2013 : 13:59:33 quote: Originally posted by someguy
Properties 1 and 3 tell you how to start forming the number in question.
Properties 1 and 3 say AB is C^2 and C=2*B. We also know C is an integer between 0 and 9. 00 (0 is 2*0) 01 (1 is not 2*1) 04 (2 is not 2*4) 09 (3 is not 2*9) 16 (4 is not 2*6) 25 (5 is not 2*5) 36 (6 is not 2*6) 49 (7 is not 2*9) 64 (8 is 2*4) 81 (9 is not 2*1)
ABC is either 000 or 648
Having the 5 digits sum to 21 rules out 000.
The number is 648DE.
The second and third digits sum to 12.
The number is 64812
Note that property 4 was not used, but is satisfied.
Nice approach someguy. I have concluded that I tend to take a longer algebraic path. But, that is what makes recreational math interesting.
From the given inforamtion: 110A + B = C^2 210D + E = B + C 3C = 2B 4E = B/2 = A/3 5Since 10A + B is a square, the only 2 digit numbers possible are 16, 25, 36, 49, 64, and 81. 6This makes the ABC possibilities 164, 255, 366, 497, 648 and 819. 7Only 648 satisfies the C = 2B requirement and therefore ABC = 648. 8WIth E = B/2 or A/3, E muat be 2 for 648D2. 9Since 10D + E = B + C, 10D + 2 = 4 + 8 making D = 1. 10Thus, the number we seek is 64,812 without using given fact #5. 11Checking: 6+4+8+1+2 = 21.
Alternatively 110A + B = C^2 210D + E = B + C 3C = 2B 4E = B/2 = A/3 5A + B + C + D + E = 21 63E + 2E + 4E + (B + C  E)/10 + E = 21 73E + 2E + 4E + 2E/10 + 4E/10  E/10 + E = 21 8Times 10, 30E + 20E + 40E + 2E + 4E  E + 10E = 210 9Collecting, 105E = 210 making E = 2. 10B = 2E making B = 4 11A = 3E making A = 6. 12With AB = 64, C = 8. 13Therefore, D = 21  6  4  8  2 = 1 for 64,812

someguy 
Posted  07/28/2013 : 13:43:23 Properties 1 and 3 tell you how to start forming the number in question.
Properties 1 and 3 say AB is C^2 and C=2*B. We also know C is an integer between 0 and 9. 00 (0 is 2*0) 01 (1 is not 2*1) 04 (2 is not 2*4) 09 (3 is not 2*9) 16 (4 is not 2*6) 25 (5 is not 2*5) 36 (6 is not 2*6) 49 (7 is not 2*9) 64 (8 is 2*4) 81 (9 is not 2*1)
ABC is either 000 or 648
Having the 5 digits sum to 21 rules out 000.
The number is 648DE.
The second and third digits sum to 12.
The number is 64812
Note that property 4 was not used, but is satisfied.


