Problem 29 · 2024 Math Kangaroo
Stretch
Number Theory
place-valuecareful-counting
A special four-digit number \(\overline{abcd}\) fulfils the equation \(\overline{abcd}=a^a+b^b+c^c+d^d\). What is the value of a?
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Answer: B — 3
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Hint 1 of 2
You need a 4-digit number equal to the sum of each digit raised to its own power.
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Hint 2 of 2
Search digit by digit: 3³ + 4⁴ + 3³ + 5⁵ lands exactly on a 4-digit number.
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Approach: each digit raised to itself
- We want abcd = aᵃ + bᵇ + cᶜ + dᵈ.
- Testing 3435: 3³ + 4⁴ + 3³ + 5⁵ = 27 + 256 + 27 + 3125 = 3435.
- So the number is 3435 and a = 3.
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