Harry Potter In Pottery Business

Number of pots he made on the first day = $x$

Total number of pots made = $x + x + 1 + x + 2....$ = $497$

Number of days = $a$

= $ax+\frac{a(a-1)}{2}$

= $\frac{2ax+a^2-a}{2} = 491$

= $a(2x+a-1) = 994$ ..... (1)

= $a(2x+a-1) = 2 \times 7 \times 71$

Solving for $x >= 0$

$a$ must be less than 32

Possible values for a = $2, 7, 71, 2 \times 7, 2 \times 71, 7 \times 71$

Values less than 31 are 2, 7, and 14

For $x$ to be minimum, $a$ must be maximum

$\therefore a = 14$

substituting in eq(1)

$a(2x+a-1) = 994$

= $14(2x+14-1) = 994$

$\implies (2x+13) = \frac{994}{14}$

$\implies x=29$

Let the number of pots made on the first day = $x$ , and the number of days in total = $y$ .

Every day, there will be $x$ pots made, plus the number of days it has been since starting - 1. So for example, on the first day, $x$ pots are made, on the second day, $x +1$ pots are made etc, up till $x + y$ pots on the final day.

So then, the total number of pots made would be:

$xy + \frac{(y)(y-1)}{2}=497$

This equation rearranges to give:

$2xy + y\times(y-1) = 994$

which multiplies out to give:

$2xy + y^2 -y = 994$

and factorises to give:

$y(2x + y -1) = 994$ .

In order for $x$ to be as small as possible, $y$ must be as large as possible.

However, $(2x + y -1) > y$ .

The factors of $994$ are $2 \times 7 \times 71$ . The factor pair which have the least difference are $14 \times 71$ . Therefore, $2x + 14 + 1 = 71$ , which simplifies to $x = \boxed{29}$

Let the number of pots he made on the first day be $\large x$ . Let the number of pots he makes on the last day be $\large y$ . So, the number of pots he makes in total is $\large y-x+1$ .

The total number of pots made is: $\large\frac{(y-x+1)(y+x)}{2}=497$

Multiplying both sides of the equation by two gives: $\large(y-x+1)(y+x)=994=2\times7\times71$

To work out the value of $x$ we subtract $(y-x+1)$ from $(y+x)$ and then add one and halve.

To get the smallest value of x we therefore want the difference between $(y-x+1)$ and $(y+x)$ to be as small as possible, with $x$ and $y$ still being positive, which is achieved by $(y-x+1)=2\times7$ and $(y+x)=71$ .

This gives $\large x=29,y=42$