perplexus.info :: Just Math : Digit addition and insertion

Digit addition and insertion (Posted on 2011-10-27)

(I) Determine all possible base ten 8-digit perfect square(s) N, with no leading zeroes, such that the number obtained by increasing the second digit (reading left to right) of N by 1 is also a perfect square.

(II)Determine all possible base ten 4-digit perfect square(s) N, with no leading zeroes, such that the number obtained by inserting two zeroes between the thousands digit and the hundreds digit of N is also a perfect square.

**** Calculators are allowed, but no computer programs.

Submitted by K Sengupta

Rating: 5.0000 (2 votes)

Solution: (Hide)

(I) 24502500 = 4950² :: 25502500 = 5050²
38564100 = 6210² :: 39564100 = 6290²
If wrap-around of digits is allowed so that 9 is incremented to 0, then:
29702500 = 5450² and 20702500 = 4550²
If, however, when you do this you also carry the 1 from what's actually a 10, you get:
99500625 = 9975² and 100500625 = 10025²
(II) 2704 = 52² :: 200704 = 448²
7569 = 87² :: 700569 = 837²

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
No Subject	K Sengupta	2023-08-05 12:31:01
supplement manufacturers	Bradley Bauer	2023-08-05 04:45:39
Mental	Bradley Bauer	2023-07-26 07:02:26
x3diamond	Bradley Bauer	2023-07-26 06:49:19
8	Bradley Bauer	2023-07-26 06:27:36
7	Bradley Bauer	2023-07-26 06:09:41
6	Bradley Bauer	2023-07-26 05:51:41
5	Bradley Bauer	2023-07-26 05:34:05
4	Bradley Bauer	2023-07-26 05:14:41
re(2): answers	Kevin	2022-01-31 10:07:01
re: answers	Charlie	2011-10-27 18:31:11
answers	Dej Mar	2011-10-27 12:32:53

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