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What is the smallest perfect square that ends with the four digits 9009?

Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Mod 7

Find the remainder when 3^{2001} is divided by 7.

Modular Fractions

Age 16 to 18 Challenge Level:

Congratulations to Wei Zhang, age 18 from Merchiston Castle School for the superb solution given below. Andrei Lazanu, age 14, School No. 205, Bucharest, Romania submitted a similar solution. \par Suppose there is a fraction $1/n$ , where $n$ belongs to $\{1,2,3,4,5,6\}$.

$$\eqalign{ 1/1: 1x &=1\ {\rm so}\ x=1 \cr 1/2: 2x &=1\ {\rm so}\ x=4 \cr 1/3: 3x &=1\ {\rm so}\ x=5 \cr 1/4: 4x &=1\ {\rm so}\ x=2 \cr 1/5: 5x &=1\ {\rm so}\ x=3 \cr 1/6: 6x &=1\ {\rm so}\ x=6 }$$

As I showed above, $1/n$ is in the set $\{1,2,3,4,5,6\}$.

Now we have $a/n = a \times 1/n$ , where $a$ belongs to $\{0,1,2,3,4,5,6\}$, this is a number from the set $\{0,1,2,3,4,5,6\}$ times a number from the set of $\{1,2,3,4,5,6\}$. Therefore we will get a set of numbers for $a/n$ , which is $\{0,1,2,3,4,5,6\}$. Solving the equation

$${1\over x} + {1\over y}= {1\over{x+y}}$$

is equivalent to solving $x^2+xy+y^2=0$.

when $x=1$ $y^2+y+1=0$ so $y^2+y=6$ $y=2$ or $y=4$
when $x=2$ $y^2+2y+4=0$ so $y^2+2y=3$ $y=1$ or $y=4$
when $x=3$ $y^2+3y+2=0$ so $y^2+3y=5$ $y=6$ or $y=5$
when $x=4$ $y^2+4y+2=0$ so $y^2+4y=5$ $y=1$ or $y=2$
when $x=5$ $y^2+5y+4=0$ so $y^2+5y=3$ $y=3$ or $y=6$
when $x=6$ $y^2+6y+1=0$ so $y^2+6y=6$ $y=3$ or $y=5$

We have found the six solution pairs $(1,2)$, $(1,4)$, $(2,4)$, $(3,6)$, $(3,5)$, $(5,6)$. The equation is symmetric in $x$ and $y$ so there are six corresponding solutions when we exchange $x$ and $y$.

We don't allow negative solutions (e.g. $x=1,\ y=-3$) because we are working entirely in the set $\{0,1,2,3,4,5,6\}$.

If we work with real numbers, we think of solving the quadratic equation $x^2+yx+y^2=0$ as an equation in $x$. The formula for the solution of this quadratic equation involves $y^2-4y^2=-3y^2< 0$ which has no real values, therefore there is no real solution for $x$.