Try this beautiful problem from Algebra based on Perfect cubes from AMC-8, 2018, Problem -25.
Perfect Cubes | AMC-8, 2018| Problem 22
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?
- 55
- 60
- 58
Key Concepts
Algebra
cube
integer
Check the Answer
But try the problem first…
Answer:$58$
AMC-8, 2018 problem 25
Challenges and Thrills in Pre College Mathematics
Try with Hints
First hint
Find the vale 0f \(2^8+1\) and \(2^{18} +1\)
Can you now finish the problem ……….
Second Hint
Find the lest and largest cubes betwwen \(2^8+1\) and \(2^{18} +1\)
can you finish the problem……..
Final Step
The value of \(2^8+1=257\)
Now less than 257 the perfect cube is \(6^3\)=216 and \(7^3=343\),which is greater than 257
Now \(2^{18}=(2^6)^3=(64)^3\) which will be the largrst cube less than \(2^{18 }+1\)
Hece the required number of cubes is 64-7+1=58
Other useful links
- https://www.cheenta.com/problem-based-on-integer-prmo-2018-problem-6/
- https://www.youtube.com/watch?v=ilbU2K2wS1I