Assembling 63964

Little Pavel was assembling building blocks (a cube is shaped like a cube). He wanted to build a big cube. However, he had 75 dice left, so he increased the edge by one die. Then he was missing 16 dice. How many cubes did he have in the kit?

Correct answer:

n =  5

Step-by-step explanation:

$$\begin{gathered} V_1=V_2=a^3=b^3 \\ {n}^3+75=(n+1{)}^3-16 \\ {n}^3+75=n^3+3n^2+3n+1-16 \\ 75=3n^2+3n+1-16 \\ 75=3n^2+3n+1-16 \\ -3n^2-3n+90=0 \\ 3n^2+3n-90=0 \\ 3...\text{ prime number} \\ 90=2\cdot 3^2\cdot 5 \\ \text{GCD}(3,3,90)=3 \\ {n}^2+n-30=0 \\ a=1;b=1;c=-30 \\ D=b^2-4ac=1^2-4\cdot 1\cdot (-30)=121 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-1\pm \sqrt{121}}{2} \\ {n}_{1,2}=\frac{-1\pm 11}{2} \\ {n}_{1,2}=-0.5\pm 5.5 \\ {n}_1=5 \\ {n}_2=-6 \\ n>0 \\ n=n_1=5 \end{gathered}$$

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