How many choices for the digit A make the number 86A40 a multiple of 8?

What choice of the digit A makes the number 135A9 have remainder 3 when divided by 9?

What is the remainder when

\[ 100 \times 101 \times 102 \times 103 \]

is divided by \(11?\)

Find the last digit of:

\[ 2222^{3141592} \]

Let \(D(n)\) equal the number of factors of an integer \(n.\) For how many values of \(n\) from \(1\) to \(300\) is \(D(n)\) odd?

How many integers less than or equal to \(255\) are relatively prime to \(255?\)

Find the smallest positive integer \(a\) that makes \(\displaystyle \frac{a}{105}\) a terminating decimal.

How many numbers from \(1\) through \(105\) have a remainder of \(2\) when divided by \(3\) and a remainder of \(3\) when divided by \(7?\)

The product of all the factors of \(156\) has a prime factorization of \(2^a b,\) where \(b\) is an odd integer.

What is the value of \(a?\)

What is the smallest four-digit number that has a remainder of \(1\) when divided by \(3,\) a remainder of \(2\) when divided by \(13,\) and a remainder of \(2\) when divided by \(17?\)