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How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit?

 May 31, 2018
 #1
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Convert 343 in base 10 to base 7

343_10 =1 000_7

 

1 000_7 / 4_7 =152 fours

1 000_7 / 5_7 =125 fives.

1 000_7 / 45_7 =13 fours and fives.

 May 31, 2018
 #2
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Write ALL numbers from 1 to 342 in base 7 as 3 digits, such as the following:

000, 001, 002, 003, 004, 005, 006, 010, 011, 012, 013, 014, 015, 016.........and so on to 666, and  1000. This gives you a total of:
3 x 49 x 7 =1,029 digits in 342 numbers in base 7. Since each number from 0 to 6 is equally present in the above 1029 digits, it, therefore, follows that there are:1,029 / 7 =147 fours and 147 fives, in any combinations of the two.

 May 31, 2018

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