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Special Numbers

August 31, 2012

Mathematics is a rich and varied landscape, full of mathematical flora and fauna. Finding your way around can be tricky at the best of times, so it’s good to have some pointers to give you a hint on which way to go next.

Of all mathematical ideas, numbers are among the simplest, so let’s look at some special numbers. I have made an Excel version of the table below, together with some other bits and pieces.

Hmm – 153, 370, 371 and 407 are calling out that all numbers are special, and so indeed all of them are (for example, these numbers are the only four with which property? See below for solution). But still…

Indeed, the numbers in the table below do appear much more often in problems (both in and out of school). Perhaps 5% to 10% of problems use such numbers, and so will be easier to solve if you know why they are special.

For example, if you see 343, almost certainly it is part of the problem because it is 7x7x7. In contrast, I don’t remember ever seeing 342 in a problem. Numbers like 342 may appear very occasionally, but for no particular reason – and therefore much, much less often.

# Powers   Exponentials Factorials* Primes
n n2 n3 2n 3n 5n n! pn

0

0

0

1

1

1

1

1

1

1

2

3

5

1

2

2

4

8

4

9

25

2

3

3

9

27

8

27

125

6

5

4

16

64

16

81

625

24

7

5

25

125

32

243

3125

120

11

6

36

216

64

729

15625

720

13

7

49

343

128

2187

78125

5040

17

8

64

512

256

6561

40320

19

9

81

729

512

19683

362880

23

10

100

1000

1024

59049

3628800

29

11

121

1331

2048

31

12

144

1728

4096

37

13

169

2197

8192

41

14

196

2744

16384

43

15

225

3375

32768

47

16

256

4096

65536

53

17

289

4913

131072

59

18

324

5832

262144

61

19

361

6859

524288

67

20

400

8000

1048576

71

21

441

73

22

484

79

23

529

83

24

576

89

25

625

97

26

676

27

729

28

784

— — Legend — —

29

841

Red = must know

30

900

Black = should know

31

961

Blue – nice to know

32

1024

Green – just for fun

* Note on factorials:
 4! = 4x3x2x1 = 24
 5! = 5x4x3x2x1 = 120
 6! = 6x5x4x3x2x1 = 720 etc

I originally made this spreadsheet for upper secondary student use, but even those who are younger (or older!) readers would benefit from familiarity with the numbers in this table.

 

 

 

 

 

 

 

 

 

 

 

153, 370, 371, 407 ?

153 = 13 + 53 + 33
370 = 33 + 73 + 03
371 = 33 + 73 + 13
407 = 43 + 03 + 73

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