kangaroo maths competition

Question

How many three-digit positive integers $ABC$ exist, such that $(A + B)^c$ is a three-digit integer and an integer power of $2$? Note: An integer power of $2$ is a number in the form $2^k$ , where $k$ is an integer.

(A) $15$

(B) $16$

(C) $18$

(D) $20$

(E) $21$

Answer 1

HINTS:

You basically have to find the number of solutions to the following equations

$$ (A+B)^C=2^7\qquad (A+B)^C=2^8\qquad (A+B)^C=2^9, \quad A,B,C\in\{0,1,2,...,9\} $$

Which directly gives some clues about what $C$ could be. Can $A$ be equal to zero?

Answer 2

Here is complete list of $21$ solutions.

• $2^7=128:$
  • $2^7=(1+1)^7\implies ABC=117$
  • $2^7=(2+0)^7\implies ABC=207$
• $2^8=4^4=16^2=256:$
  • $2^8=(1+1)^8\implies ABC=118$
  • $2^8=(2+0)^8\implies ABC=208$
  • $4^4=(1+3)^4\implies ABC=134$
  • $4^4=(2+2)^4\implies ABC=224$
  • $4^4=(3+1)^4\implies ABC=314$
  • $4^4=(4+0)^4\implies ABC=404$
  • $16^2=(7+9)^2\implies ABC=792$
  • $16^2=(8+8)^2\implies ABC=882$
  • $16^2=(9+7)^2\implies ABC=972$
• $2^9=8^3=512:$
  • $2^9=(1+1)^9\implies ABC=119$
  • $2^9=(2+0)^9\implies ABC=209$
  • $8^3=(1+7)^3\implies ABC=173$
  • $8^3=(2+6)^3\implies ABC=263$
  • $8^3=(3+5)^3\implies ABC=353$
  • $8^3=(4+4)^3\implies ABC=443$
  • $8^3=(5+3)^3\implies ABC=533$
  • $8^3=(6+2)^3\implies ABC=623$
  • $8^3=(7+1)^3\implies ABC=713$
  • $8^3=(8+0)^3\implies ABC=803$