Posted - 07/08/2013 : 15:44:09 The digits of a 5 digit number ABCDE have the following relationships:

1--The first two digits are the square of the third digit. 2--The fourth and fifth digits are the sum of the second and third digits. 3--The third digit is twice the second digit. 4--The fifth digit is one half the second digit and one third the first digit. 5--The sum of the five digits is 21. 6--What 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: 1--10A + B = C^2 2--10D + E = B + C 3--C = 2B 4--E = B/2 = A/3 5--Since 10A + B is a square, the only 2 digit numbers possible are 16, 25, 36, 49, 64, and 81. 6--This makes the ABC possibilities 164, 255, 366, 497, 648 and 819. 7--Only 648 satisfies the C = 2B requirement and therefore ABC = 648. 8--WIth E = B/2 or A/3, E muat be 2 for 648D2. 9--Since 10D + E = B + C, 10D + 2 = 4 + 8 making D = 1. 10--Thus, the number we seek is 64,812 without using given fact #5. 11--Checking: 6+4+8+1+2 = 21.

Alternatively 1--10A + B = C^2 2--10D + E = B + C 3--C = 2B 4--E = B/2 = A/3 5--A + B + C + D + E = 21 6--3E + 2E + 4E + (B + C - E)/10 + E = 21 7--3E + 2E + 4E + 2E/10 + 4E/10 - E/10 + E = 21 8--Times 10, 30E + 20E + 40E + 2E + 4E - E + 10E = 210 9--Collecting, 105E = 210 making E = 2. 10--B = 2E making B = 4 11--A = 3E making A = 6. 12--With AB = 64, C = 8. 13--Therefore, 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.