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.

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

Keep Smiling
Malay

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: $N \:=\:aaaa_6 \:=\:6^3a + 6^2a + 6a + 6 \:=\:259a$

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

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

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

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