1. ## Converting to Bases

"A number X is converted to base 7 and becomes a four-digit number. Its leftmost digit is removed and written against as the rightmost digit. The number thus obtained is twice X. Find the decimal representations of all such numbers X."

Any help on this is much appreciated, I've been trying to figure it out all week.

2. Originally Posted by ilove22
"A number X is converted to base 7 and becomes a four-digit number. Its leftmost digit is removed and written against as the rightmost digit. The number thus obtained is twice X. Find the decimal representations of all such numbers X."

Any help on this is much appreciated, I've been trying to figure it out all week.
Put:

X = A + B*7 + C* 7^2 + D*7^3

with 0<= A,B,C <=6, 1<=D<=6

Then:

Y = D + A*7 + B*7^2 + C*7^3

and we are told that Y=2*X.

So we may conclude that 2*D=C (there is no base 7 carry), so
D<=3, but then 2A=D, so D is even, and as it is the modt significant
digit (base 7) of a four digit number it !=0, so D=2. Then A=1, and C=4.

So we now have:

X = 1 + B*7 + 4* 7^2 + 2*7^3
Y = 2 + 1*7 + B*7^2 + 4*7^3

So 2*B=1 or 8, but neither of these is possible, so there is no such number
X with the given properties!

Have you got the question right?
Can you see a hole in my argument?

RonL