## SUBJECT I

Let the sets $\mathrm{A}=\left\{\left.\frac{a}{b} \right\rvert\, a, b \in \mathrm{N}^{*}, a+b=2015\right\}$ and $\mathrm{B}=\left\{\mathrm{x} \in \mathbf{Q} \left\lvert\, \mathrm{x}=\frac{\mathrm{a}}{\mathrm{b}}\right., \mathrm{a} \in \mathrm{D}_{12}, \mathrm{~b} \in \mathrm{D}_{6}\right\}$. Calculate:

a) the product of all elements of set A;

b) the product of all elements of set B.
## SUBJECT I

a) The fractions are of the form $\frac{k}{2015-k}$, where $k$ takes values from 1 to 2014 ................................1p

The product of the numerators is $1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2014$ as is the product of the denominators.......................................1p

The product of the fractions is 1 ...................................................................................................1p

b) The divisors of 12 are: 1, 2, 3, 4, 6, 12 and those of 6 are: 1, 2, $3,6 \ldots \ldots . . . . . . . . . . . . . . . . . . . . . .1 p$

All combinations are considered, which are 24 in number because each divisor of 12 appears 4 times, and each divisor of 6 appears 6 times.

Equivalent fractions are eliminated............. ......................................................................1p

Finalization, $2^{6}$ is obtained................................................................................................... $2 p$

TOTAL ......................................................................................................................... 7p