## 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
$328880160 \; = \; 237 \; \times \; 1387680$

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.