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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

But try the problem first…

Answer:$58$

Source

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