Representation of a number in powers of other

Given two numbers w and m, we need to determine whether it is possible to represent m in terms of powers of w. The powers of number w can be added or subtracted to obtain m and each powers of w can be used only once .

Examples:

```Input : 3 7
Output : Yes
As 7 = 9 - 3 + 1 (3^2 - 3^1 + 3^0 )
so it is possible .

Input : 100 50
Output : No
As 50 is less than 100 so we can never
represent it in the powers of 100 .```

Here we have to represent m in terms of powers of w used only once so it can be shown through the following equation .
c0 + c1*w^1 + c2*w^2 + … = m —— (Equation 1)

Where each c0, c1, c2 … are either -1 (for subtracting that power of w ), 0 (not using that power of w ), 1 (for adding that power of w ) .

=> c1*w^1 + c2*w^2 + … = m – c0
=> w(c1 + c2*w^1 + c3*w^2 + … ) = m – c0
=> c1 + c2*w^1 + c3*w^2 + … = (m – c0)/w —— (Equation 2)

Now, notice equation 1 and equation 2 — we are trying to solve the same problem all over again. So we have to recurse till m > 0 . For such a solution to exist (m — ci) must be a multiple of w, where ci is the coefficient of the equation . The ci can be -1, 0, 1 . So we have to check for all three possibilities ( ( m – 1 ) % w == 0), ( ( m + 1 ) % w == 0) and ( m % w == 0) . If it is not, then there will not be any solution.

`// CPP program to check if m can be represented`
`// as powers of w.`
`#include <bits/stdc++.h>`
`using` `namespace` `std;`
`bool` `asPowerSum(``int` `w, ``int` `m)`
`{`
`    ``while` `(m) {`
`        ``if` `((m - 1) % w == 0) `
`            ``m = (m - 1) / w;`
`       ``else` `if` `((m + 1) % w == 0) `
`            ``m = (m + 1) / w;`
`        `
`        ``else` `if` `(m % w == 0) `
`            ``m = m / w;`
`        `
`        ``else`
`            ``break``; ``// None of 3 worked.`
`    ``}`
`    ``// If m is not zero means, it can't be `
`    ``// represented in terms of powers of w.`
`    ``return` `(m == 0);`
`}`
`// Driver code`
`int` `main()`
`{`
`    ``int` `w = 3, m = 7;`
`    ``if` `(asPowerSum(w, m))`
`        ``cout << ``"Yes"` `<< endl;`
`    ``else`
`        ``cout << ``"No"` `<< endl;`
`   ``return` `0;`
`}`

Output:

```Yes

```

Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.
source and credits:http://www.geeksforgeeks.org/representation-number-powers/
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rakesh