if you check out each number out each number's posiblities and put it into a formula, you will obtain: for the number 2: (x is any natural number, and it's diferent for every equation) 1+ 2x Number 3 29+30x Number 4 19+ 20x Number 5 9+ 10x Number 6 29 + 30x Number 7 69 + 70x Number 8 39 + 40x Number 9 89+90x Number 10 9 + 10X So we need a number that is equivalent to all those equations. we have: 1+2X 29+30X 19+20X 9+10X 69+70X 39+40x 89+90X So, we go and analyse the equations who give bigger products for the smallest number of X. those are: 89+90x and 69 + 70x. Replacing one of the x for another letter (y) we get: 89+90y = 69 + 70x <=> 20 = 70x - 90y With that, we can now replace the "x" and the "y". for that formula to work, after some calculations, I notice that the x must go for 8 + 9z and the y must go 6+7z So, starting for the 1st, if z = 0, x=8 and y=6. we replace them in the equation 89 + 90 x 6 = 629. However, this number doesnt exist on the equation 39+40x, so its invalid. You keep going, z=1; x= 17 and y = 13, the final =1259, also doesnt exist on the equation 39+40x, so its invalid. z=2, the final is 1889, also invalid. Then, finally, z=43 y = 6+(7x3) =27........ 89 + 90x27 = 2519; which is valid for all the equations.