From base 10 to base 4, then in base 7 to base 10

 
 
N \; base-10 \; = \; x \; base-4,    then   x \; base-7 \; = \; M \; base-10

such that   M   is a multiple of   N

 

for example,

 

N \; = \; 33

33 \; base-10 \; = \; 201 \; base-4
then
201 \; base-7 \; = \; 99 \; base-10

99 \; = \; 3 \; \times \; 33

here’s another example:

N \; = \; 219

219 \; base-10 \; = \; 3123 \; base-4
3123 \; base-7 = \; 1095 \; base-10
1095 \; = \; 5 \; \times \; 219

 

Find other examples.
 

 
840   base-10   =   31020   base-4
31020   base-7   =   7560   base-10
7560 \; = \; 9 \; \times \; 840

1055   base-10   =   100133   base-4
100133   base-7   =   16880   base-10
16880 \; = \; 16 \; \times \; 1055

4220   base-10   =   1001330   base-4
1001330   base-7   =   118160   base-10
118160 \; = \; 28 \; \times \; 4220

16880   base-10   =   10013300   base-4
10013300   base-7   =   827120   base-10
827120 \; = \; 49 \; \times \; 16880

95322   base-10   =   113101122   base-4
113101122   base-7   =   6958506   base-10
6958506 \; = \; 73 \; \times \; 95322

178500   base-10   =   223211010   base-4
223211010   base-7   =   13566000   base-10
13566000 \; = \; 76 \; \times \; 178500

714000   base-10   =   2232110100   base-4
2232110100   base-7   =   94962000   base-10
94962000 \; = \; 133 \; \times \; 714000

981540   base-10   =   3233220210   base-4
3233220210   base-7   =   135452520   base-10
135452520 \; = \; 138 \; \times \; 981540

1387680   base-10   =   11102302200   base-4
11102302200   base-7   =   328880160   base-10
328880160 \; = \; 237 \; \times \; 1387680

 
 
 
 
 

 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , , . Bookmark the permalink.

2 Responses to From base 10 to base 4, then in base 7 to base 10

  1. paul says:

    Just using the last line we have

    7560 = 9 x 840
    16880 = 16 x 1055
    118160 = 28 x 4220
    827120 = 49 x 16880
    6958506 = 73 x 95322
    13566000 = 76 x 178500
    94962000 = 133 x 714000
    135452520 = 138 x 981540

    Paul.

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