Prove by induction : show that $24 or more can be dispensed from an ATM using only $5 and $7 notes.
how do i go about doing this?
1st can you do this with \$24?
$2\times 5 + 2 \times 7 = 24$, so yes you can.
Suppose you can do it with \$D. Show you can do it with \$(D+1).
$D = 5m+7k,~~m,k \in \mathbb{N}$
$D+1 = 5m + 7k + 1 = \dots$
you should be able to finish from here.
There are a couple different ways of obtaining D+1 and you'll need to use both of them.
I would use five base cases:
$\displaystyle 24 = 2\times 5 + 2\times 7$
$\displaystyle 25 = 5\times 5+ 0 \times 7$
$\displaystyle 26 = 1\times 5 + 3\times 7$
$\displaystyle 27 = 4\times 5 + 1\times 7$
$\displaystyle 28 = 0\times 5 + 4\times 7$
For any $\displaystyle n\ge 24$, you can write $\displaystyle n=5k+24, n=5k+25, n=5k+26, n=5k+27,\text{ or }n=5k+28$ for some nonnegative integer $\displaystyle k$. It can only be written as one of them. Depending on how it can be written, you use the appropriate breakdown of the base number, then add $\displaystyle k$ $\$5$ notes.