Two-digit number puzzle

In a two-digit number, the number of tens is three more than the number of ones. If we multiply the original number by a number written with the same digits but in the reverse order, we get the product 3 478. Determine the actual number.

Final Answer:

x =  74

Step-by-step explanation:

$$\begin{gathered} a=3+b \\ (a\cdot 10+b)\cdot (a+10\cdot b)=3478 \\ (a\cdot 10+(a-3))\cdot (a+10\cdot (a-3))=3478 \\ (a\cdot 10+(a-3))\cdot (a+10\cdot (a-3))=3478 \\ 121a^2-363a-3388=0 \\ 121=11^2 \\ 363=3\cdot 11^2 \\ 3388=2^2\cdot 7\cdot 11^2 \\ \text{GCD}(121,363,3388)=11^2=121 \\ {a}^2-3a-28=0 \\ p=1;q=-3;r=-28 \\ D=q^2-4pr=3^2-4\cdot 1\cdot (-28)=121 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{3\pm \sqrt{121}}{2} \\ {a}_{1,2}=\frac{3\pm 11}{2} \\ {a}_{1,2}=1.5\pm 5.5 \\ {a}_1=7 \\ {a}_2=-4 \\ a>0 \\ a=a_1=7 \\ b=a-3=7-3=4 \\ x=a\cdot 10+b=7\cdot 10+4=74 \end{gathered}$$

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