Problem 3 · Five integers and a power
Trying small bases: $2^{6} = (3\cdot 5 + 1)\cdot 4$.
Integer answer, at most 4 digitsFind 5 integers such that $0 < a < b < c < d < e < 10$ and $a^{e} = (bd + 1) \cdot c$. (As your answer, give the product of these five numbers.)
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: $a$ cannot be $1$ ($1^{e} = 1$ is too small) and if $a \ge 3$ the power $a^{e}$ with $e > 4$ becomes huge. Try $a = 2$.
With $a = 2$ and moderate exponents: $2^{6} = 64$. We look for $b, c, d$ with $2 < b < c < d < 6$ and $(bd+1)c = 64$: trying $b=3$, $c=4$, $d=5$:
$$(3 \cdot 5 + 1) \cdot 4 = 16 \cdot 4 = 64 = 2^{6}. \;\checkmark$$
Hence $(a,b,c,d,e) = (2,3,4,5,6)$ — five consecutive integers! The product:
$$2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720.$$
Answer: 720