## Wednesday, 4 July 2012

### repdigit endings to squares

James Tanton @jamestanton Wrote:

"2^2 ends with 4 and 12^2 ends with 44. Is there a square than ends 444? How about one that ends 4444?"

To which I think the answer is yes,..... and no.

38 squared is 1444

462 squared is 213444

538 squared is 289444 and there are many more... (so I'm pretty firm on the "yes".)

but I don't think you will ever see a square ending in 4444. Here's why:

This is a list of the square numbers that end in 444 from 1 to 100,000,000

00001444

00213444

00289444

00925444

01077444

02137444

02365444

03849444

04153444

06061444

06441444

08773444

09229444

11985444

12517444

15697444

16305444

19909444

20593444

24621444

25381444

29833444

30669444

35545444

36457444

41757444

42745444

48469444

49533444

55681444

56821444

63393444

64609444

71605444

72897444

80317444

81685444

89529444

90973444

99241444

Notice a pattern? Look at that digit in the thousands place...

Those are the squares of these numbers:

0038

0462

0538

0962

1038

1462

1538

1962

2038

2462

2538

2962

3038

3462

3538

3962

4038

4462

4538

4962

5038

5462

5538

5962

6038

6462

6538

6962

7038

7462

7538

7962

8038

8462

8538

8962

9038

9462

9538

9962

and another pattern appears, that may help us with a more formal proof:

I hadn't thought too much about the idea of digit repeats at the end of squares, and maybe you haven't either, so here is a quick question to get you started.

When speaking of square numbers:

What ending digits can repeat 2, 3 or more times? More particularly, is there any ending digit that repeats four (or more) times?

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