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.

You are watching: Convert base 10 to base 5

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**:

$$\beginalign*\frac7275=145+\frac25\longrightarrow& \colorred2\\\frac1455=29\longrightarrow&\colorred0\\\frac295=5+\frac45\longrightarrow&\colorred4\\\frac55=1\longrightarrow&\colorred0\\\frac15=\frac15\longrightarrow&\colorred1\endalign*$$

Now put them in inverse order and voila!

$$727_10=10402_5$$

Can you watch why that works?

You need to keep separating by $5$ and also write every the remainders.

So

$$727 = 5\cdot 145 + 2$$$$145 = 5\cdot29 + 0$$$$29 = 5\cdot5 + 4$$$$5 = 5\cdot1 + 0$$$$1 = 5\cdot0 + 1$$

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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