Matemakak 9432

The cookbook by Matěj Matemakak says:
The greatest common divisor of flour weight and sugar weight is 15, the greatest common divisor of sugar weight and lemon peel weight is 6, the product of sugar weight and lemon peel weight is 1800, and the smallest common multiple of flour weight and sugar weight is 3150.
What are flour, sugar, and lemon peel weights in a lemon pie recipe?

Correct answer:

m =  315 g
c =  150 g
k =  12 g

Step-by-step explanation:

$$\begin{gathered} GCD(m,c)=15 \\ GCD(c,k)=6 \\ ck=1800 \\ LCM(m,c)=3150 \\ ck=1800 \\ {c}_1\cdot 6\cdot k_1\cdot 6=1800 \\ {c}_1\cdot k_1=50 \\ {c}_1\cdot k_1=2\cdot 5\cdot 5 \\ {c}_1=f{r}_2\frac{5}{2}\cdot 5|5\cdot 5 \\ {c}_1=5\cdot 5=25 \\ c=6\cdot c_1=6\cdot 25=150\text{g} \\ LCM(m,c)=3150 \\ LCM(m,150)=3150 \\ 150=2\times 3\times 5\times 5 \\ 3150=2\times 3\times 3\times 5\times 5\times 7 \\ LCM(m,2\times 3\times 5\times 5)=2\times 3\times 3\times 5\times 5\times 7 \\ m=5\cdot 3\cdot 3\cdot 7=315\text{g} \end{gathered}$$

$c=150=150\text{g}$

$$\begin{gathered} k=1800/c=1800/150=12=12\text{g} \\ \text{ Verifying Solution: } \\ 315=3^2\cdot 5\cdot 7 \\ 150=2\cdot 3\cdot 5^2 \\ \text{GCD}(315,150)=3\cdot 5=15 \\ {t}_1=GCD(m,c)=GCD(315,150)=15 \\ 150=2\cdot 3\cdot 5^2 \\ 12=2^2\cdot 3 \\ \text{GCD}(150,12)=2\cdot 3=6 \\ {t}_2=GCD(c,k)=GCD(150,12)=6 \\ 315=3^2\cdot 5\cdot 7 \\ 150=2\cdot 3\cdot 5^2 \\ LCM(315,150)=2\cdot 3^2\cdot 5^2\cdot 7=3150 \\ {t}_3=LCM(m,c)=LCM(315,150)=3150 \end{gathered}$$

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