Number theory – Find the number satisfying the condition.

Question

Find a 3 digit number which equals to 4 times the product of its digits.

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My approach:

I considered the Number to be $\overline{ABC}$ then wrote the relation $$100A+10B+C=4A\times B\times C$$ But i don't know how to proceed further. Please help.

Answer 1

Hint: One good way to begin with is always to look at congruence class. For example it's easy to see that $4$ divides the RHS, hence it has to divide the LHS and in particular $4$ divides $c + 10b$. Thus $c$ is even which implies even a better condition: $8$ divides $100a + 10b + c$.

Another intresting condition is that if any digits were $5$ then your number is divisible by $10$ and so $c = 0$, but the RHS then implies that your number is $0$.

[from Barry Cipra observation]

Checking the equation mod $3$ we get $$ A + B + C = ABC \mod 3 $$ Assuming that $A,B,C$ are all non divisible by $3$, then we are left with 4 cases to check, but modulus $3$ they are all not possibile $$ \begin{array}{ll} 1 + 1 + 1 = 0 \neq 1 = 1 \cdot 1 \cdot 1 & \quad 1 + 1 + 2 = 1 \neq 2 = 1 \cdot 1 \cdot 2 \\ 1 + 2 + 2 = 2 \neq 1 = 1 \cdot 2 \cdot 2 & \quad 2 + 2 + 2 = 0 \neq 2 = 2 \cdot 2 \cdot 2 \end{array} $$

I encourage you to find more conditions, until eventually you are left with a feasible number of cases that can be checked by hand