a number x is converted to base 7 and becomes a four digit number.?

Question

its leftmost digit is removed and written again as the rightmost digit. the number thus obtained is twice x. find the decimal representations of all such numbers x

thanks guys

Answer 1

I made a program that converted all integers from 343 to 2400 into base 7 (these are all the base 10 integers that become four-digit numbers in base 7). It took the left-most digit from each base 7 representation and stuck it on the right side, then converted this new base 7 number back to base 10. Then it checked to see if the new base 10 number was exactly twice the initial number. If it was, the program prints out the initial number.

I get exactly two results: 480 and 960

480 base 10 = 1254 base 7
1254 ==> 2541
2541 base 7 = 960 base 10
960 = 2*480

960 base 10 = 2541 base 7
2541 ==> 5412
5412 base 7 = 1920 base 10
1920 = 2*960

Answer 2

1000 base 7 is the smallest 4 digit number in base 7
6666 base 7 is the biggest 4 digit number
1000 base 7 is 1 x 7^3 = 7*7*7 = 49 *7 = 343
6666 base 7 is 6*343 +6*49 +6*7 +6
= 2058 + 294 + 42 +6
=2400
or 1 less that 7^4 oops lol could have saved some arithmetic there

so any number between 343 and 2400 inclusive will be a 4 digit number base 7.

xyy *2 = xyyx or xyy *2 = yyx thats a bit unclear

Answer 3

I'm confused. The "number thus obtained is twice x".... Does that mean that the number in base 7 is in digits twice the original number in base 10 OR that the number thus obtained (converted to base 10) is twice the original number.

Answer 4

each and each set of weights acts like 2 digits of binary. The a million-gram weights can degree 0 to 3 grams, in basic terms like the two staggering-maximum binary digits. The 4-gram weights can degree 0, 4, 8, or 12 grams, in basic terms like the subsequent 2 binary digits. together they are able to function as much as any value between 0 and 15. The sixteen-gram weights characterize the subsequent 2 binary digits etc... With a entire of 10 binary digits (2^10 = 1024), any value between 0 and 1023 (1024 entire values) could properly be represented.