250 Answers- DIGITAL COINCIDENCES
If we multiply 497 by 2 we get the product 994. If we add together 497 and
2 we get 499. The figures are the same in both cases. Also 263 multiplied by 2
and added to 2 will give 526 and 265 respectively.
[Harry Lindgren points out that by inserting 9's one obtains two answers
with any desired number of digits: 4997 + 2 = 4999; 4997 X 2 = 9994;
2963 + 2 = 2965; 2963 X 2 = 5926; and similarly for 49997 + or X 2;
29963 + or X 2; and so on.-M. G.]
- PALINDROMIC SQUARE NUMBERS
The square of 836 is 698896, which contains an even number of digits and
reads backwards and forwards alike. There is no smaller square number con-
taining an even number of figures that is a palindrome.- FACTORIZING
If the number of noughts enclosed by the two ones is 2 added to any mul-
tiple of 3, two factors can always be written down at once in this curious way.
1001 = 11 X 91; 1000001 = 101 X 9901; 1000000001 = 1001 X 999001;
1000000000001 = 10001 X 99990001. The last is our required answer, and
10001 = 73 X 137. The multiple of 3 in 11 is 3: therefore we insert 3 noughts
in each factor and one more 9.
If our number contained 10 1 noughts, as I suggested, then the multiple of 3
is 33 and the factors will contain 33 noughts and 34 nines in the form shown.
If the number of noughts in the number be even you can get two factors in
this way: 1001 = 11 X 91; 100001 = 11 X 9091; 10000001 = 11 X 909091,
and so on.
- FIND THE FACTORS
The factors of 1234567890 are 2 X 3 X 3 X 5 X 3607 X 3803. If we mul-
tiply 3607 by 10 and 3803 by 9 we get two composite factors 36070 and
34227, which multiplied together produce 1234567890 and have the least pos-
sible difference between them.