Find the remainder when 21^42 divided by 441.

Given:

Number = 21⁴²

Divisor = 441

Formula used:

If a number N can be expressed as k × m, where k is an integer and m is the divisor, then the remainder when N is divided by m is 0.

Calculations:

We are asked to find the remainder when 21⁴² is divided by 441.

We can write the divisor as 441 = 21 × 21.

Now, let's look at the number 21⁴².

21⁴² = 21 × 21 × 21 × ... × 21 (42 times)

We can group the first two terms:

21⁴² = (21 × 21) × (21 × 21 × ... × 21) (40 times)

Since 21 × 21 = 441,

21⁴² = 441 × (21⁴⁰)

Let k = 21⁴⁰. Since 21 is an integer, k is also an integer.

So, 21⁴² can be expressed in the form of k × 441, where k is an integer.

If a number can be expressed as a multiple of the divisor, the remainder is 0.

∴ The remainder when 21⁴² is divided by 441 is 0.