When a positive integer N is written in base 6 , it is a four digit number such that all its four digits are the same. When N is written in base ten, it is a three digit number such that all its three digits are equal. Find the decimal representations of all such numbers N.

2. Originally Posted by fring
When a positive integer N is written in base 6 , it is a four digit number such that all its four digits are the same. When N is written in base ten, it is a three digit number such that all its three digits are equal. Find the decimal representations of all such numbers N.

$\displaystyle (aaaa)_6=(bbb)_{10}$
$\displaystyle (216+36+6+1)a=(100+10+1)b$
$\displaystyle 259a=111b$
$\displaystyle 7a=3b$
$\displaystyle a=3,b=7$

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3. Hello, fring!

I will assume that you know something about number bases.
Otherwise, you shouldn't be trying problems like this . . .

When a positive integer N is written in base 6 , it is a 4-digit number such that all its digits are equal.
When N is written in base ten, it is a 3-digit number such that all its digits are equal.
Find the decimal representations of all such numbers N.

In base-6: $\displaystyle N \:=\:aaaa_6 \:=\:6^3a + 6^2a + 6a + 6 \:=\:259a$

In base-10: $\displaystyle N \:=\:bbb_{10} \:=\:10^2b + 10b + b \:= \:111b$

Hence: .$\displaystyle 259a \:= \:111b\quad\Rightarrow\quad 7a = 3b$

Since $\displaystyle a$ is a digit in base-6 and $\displaystyle b$ is a digit in base-10,
. . the only solution is: $\displaystyle a = 3,\;b = 7$

Therefore: .$\displaystyle N \:=\:3333_6 \:= \:777_{10}$