Sum Power Cube F00B3D

1. **State the problem:** We want to express $155$ as the sum of a power of $2$ and a cube number. 2. **Recall definitions:** - A power of $2$ is any number of the form $2^n$ where $n$ is a non-negative integer. - A cube number is any number of the form $m^3$ where $m$ is an integer. 3. **Approach:** We look for integers $n$ and $m$ such that: $$155 = 2^n + m^3$$ 4. **Try possible cube numbers less than $155$:** - $1^3 = 1$ - $2^3 = 8$ - $3^3 = 27$ - $4^3 = 64$ - $5^3 = 125$ - $6^3 = 216$ (too large) 5. **Check each cube and see if $155 - m^3$ is a power of 2:** - For $m=5$, $155 - 125 = 30$, which is not a power of 2. - For $m=4$, $155 - 64 = 91$, not a power of 2. - For $m=3$, $155 - 27 = 128$, which is $2^7$. 6. **Conclusion:** $$155 = 2^7 + 3^3$$ This shows $155$ can be expressed as the sum of a power of 2 and a cube number. **Final answer:** $$155 = 128 + 27 = 2^7 + 3^3$$