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## On the Interesting Properties of the Number 37

I first noticed these patterns while playing around with a calculator as a child.
I still do this occasionally, so this list has expanded over time.

### Divisibility

111/37 = 3

222/37 = 6

333/37 = 9

444/37 = 12

555/37 = 15

666/37 = 18

777/37 = 21

888/37 = 24

999/37 = 27

Also notice that the sum of the digits adds up to the result (e.g. 1+1+1 = 3).

### Interactions with Other Primes

The sum of the digits in the product yields the original number:

11*37 = 407

4+0+7 = 11

13*37 = 481

4+8+1 = 13

17*37 = 629

6+2+9 = 17

For single digit primes (except for 3, see the triplets above) this property can be seen by summing the digits twice:

2*37 = 74

7+4 = 11

1+1 = 2

5*37 = 185

1+8+5 = 14

1+4 = 5

7*37 = 259

2+5+9 = 16

1+6 = 7

The sum of the digits in the product multiplied by 37 again produces a permutation of the original product:

23*37 = 851

8+5+1 = 14

14*37 = 518

The sum of the digits in the product is the reverse of the original number.
Notice that this works with all two digit primes ending in 1:

11*37 = 407

4+0+7 = 11

31*37 = 1147

1+1+4+7 = 13

41*37 = 1517

1+5+1+7 = 14

61*37 = 2257

2+2+5+7 = 16

71*37 = 2627

2+6+2+7 = 17

91*37 = 3367

3+3+6+7 = 19

The sum of the digits of the product equals the sum of the digits of the original number:

19*37 = 703

7+0+3 = 1+9

29*37 = 1073

1+0+7+3 = 2+9

79*37 = 2923

2+9+2+3 = 7+9

83*37 = 3071

3+0+7+1 = 8+3

89*37 = 3293

3+2+9+3 = 8+9

This property also holds for some of the multiples of these primes, for example:

19*5 = 95

95*37 = 3515

3+5+1+5 = 9+5

29*2 = 58

58*37 = 2146

2+1+4+6 = 5+8

19*37 = 703

703*37 = 26011

2+6+0+1+1 = 7+0+3