I am gaining the exact same number as soon as doing so: $7\times 10^2 + 2\times10^1+7\times10^0 = 727$, nothing changes.
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Help me number it out!
You have to count in regards to $5^0, 5^1, 5^2, 5^3$ and $5^4$, no in term of $100 ,10$ and also $1$.
Start by the greatest power that $5$ smaller than her number, i.e. $5^4=625$ here, and check just how much you deserve to multiply it there is no exceding (this will certainly be a number in $1,2,3,4$).
Take the leftover (base 10, 727-625=102) and repeat till you with $0$.
You must have$$\beginalign727_10&= 625+100+2\\&= 1\times 5^4+4\times 5^2+2\times 5^0 \\&= 10402_5.\endalign$$
Trick: divide consecutively the quotients by $\;5\;$ and keep beside the residues :
Now put them in inverse order and voila!
Can you watch why that works?
You need to keep separating by $5$ and also write every the remainders.
$$727 = 5\cdot 145 + 2$$$$145 = 5\cdot29 + 0$$$$29 = 5\cdot5 + 4$$$$5 = 5\cdot1 + 0$$$$1 = 5\cdot0 + 1$$
Write all the remainders in reverse: $10402$
And ta-da! there is your answer :-)
The trick is to establish that\beginalign*727 &= 625 + 0*125 + 4*25 + 0*5 + 2 \\ &= 5^4 + 0 *5^3 + 4*5^2 + 0*5^1 + 2*5^0.\endalign*So the price is $10402$.
You need to divide every number by 5 :
$$\beginaligned727 &= 5 \cdot 145 &+ \colorred2\\145 &= 5 \cdot 29 &+ \colorgreen0\\29 &= 5 \cdot 5 &+ \colorblue4\\5 &= 5 \cdot 1 &+ \colormagenta0\\1 &= 5 \cdot 0 &+ \colorbrown1\endaligned$$
Hence the prize is $\colorbrown1\colormagenta0\colorblue4\colorgreen0\colorred2_5 = 1 \cdot 5^4 + 0 \cdot 5^3 + 4 \cdot 5^2 + 0 \cdot 5^1 + 2 \cdot 5^0 = 727$
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