Find a base 7 three-digit number which has its digits reversed when expressed in base 9. (You do not need to indicate the base with a subscript for this answer.)

Mellie
Jun 7, 2015

#2**+10 **

Let the number be XYZ XY and Z are all 1 digit numbers between 0 and 6 inclusive

X and Z cannot by 0

Z must be smaller than X

So Y could possible be 0,12,3,4,5,6

So X could possible be 2,3,4,5,6

So Z could possible be 1,2,3,4,5

$$\\X*7^2+Y*7+Z=Z*9^2+Y*9+X\\\\

49X+7Y+Z=81Z+9Y+X\\\\$$

I might just use trial and error Mellie

**Try Z=1**

49X+7Y+1=81*1+9Y+X

48X-2Y=80

24X-Y= 40 Multiples of 24 are 24,48, Neither of these - an allowed y =40

so Z is not 1

**Try Z=2**

49X+7Y+2=81*2+9Y+X

48X-2Y=160

24X-Y=80 Multiples of 24 are 24,48,72,96 Neither of these - an allowed y =80

so Z is not 2

**Try Z=3**

49X+7Y+3=81*3+9Y+X

48X-2Y=240

24X-Y=120 The first multiple of 24 that is BIGGER than or equal to 120 is 120

SO Z can be 3, X=5 and Y=0

**$$\\\mathbf{503_7=305_9}$$ **

**check**

$${\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{49}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{248}}$$

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{81}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{248}}$$ (Edited: Thanks Chris )

There might be a much quicker way of doing this

Melody
Jun 8, 2015

#2**+10 **

Best Answer

Let the number be XYZ XY and Z are all 1 digit numbers between 0 and 6 inclusive

X and Z cannot by 0

Z must be smaller than X

So Y could possible be 0,12,3,4,5,6

So X could possible be 2,3,4,5,6

So Z could possible be 1,2,3,4,5

$$\\X*7^2+Y*7+Z=Z*9^2+Y*9+X\\\\

49X+7Y+Z=81Z+9Y+X\\\\$$

I might just use trial and error Mellie

**Try Z=1**

49X+7Y+1=81*1+9Y+X

48X-2Y=80

24X-Y= 40 Multiples of 24 are 24,48, Neither of these - an allowed y =40

so Z is not 1

**Try Z=2**

49X+7Y+2=81*2+9Y+X

48X-2Y=160

24X-Y=80 Multiples of 24 are 24,48,72,96 Neither of these - an allowed y =80

so Z is not 2

**Try Z=3**

49X+7Y+3=81*3+9Y+X

48X-2Y=240

24X-Y=120 The first multiple of 24 that is BIGGER than or equal to 120 is 120

SO Z can be 3, X=5 and Y=0

**$$\\\mathbf{503_7=305_9}$$ **

**check**

$${\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{49}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{248}}$$

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{81}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{248}}$$ (Edited: Thanks Chris )

There might be a much quicker way of doing this

Melody
Jun 8, 2015