Indices

Add powers of 3 and powers of 7 and get multiples of 11.

What is the units digit for the number 123^(456) ?

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

What are the possible remainders when the 100-th power of an integer is divided by 125?

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

What are the last two digits of 2^(2^2003)?

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.