Knowledge Check

What is the product of all the positive integer factors of 199?

How many of the factors of 900 have exactly 18 factors?

The digits of a 5-digit positive integer are 1’s, 2’s, and 3’s with at least one of each. What is the smallest such integer that is divisible by both 8 and 9?

What is the remainder when:

Is divided by 9?

How many positive integers less than 10,000 are divisible by all of the following: 2, 3, 4, 5, 6, 8, 9, 10, and 11?

What is the base-10 value of the greatest five-digit base-4 integer?

Five- hundred people stand in a circle. Starting with Roy and working to his left, each person counts off a number from 1 through 6 and then starting over again (1, 2, 3, 4, 5, 6, 1, 2, 3,…) until everyone has counted a number. What number is counted off by the person standing to Roy’s right?

The number 222 is raised to the 222nd power and then multiplied by 9. What is the units digit of the result?

Convert:

To a fraction in simplest form.

In the prime factorizations of integers a, b, and c: a and b have exactly 4(not necessarily distinct) prime factors in common, a and c have exactly 5 prime factors in common, while b and c share 6. What is the least positive product abc for three positive integers which satisfy these conditions?

For how many positive values of n where n <= 100 is 1/n represented by a terminating decimal?

What base-10 fraction is represented by the base-5 decimal:

Add:

Express the result as a fraction in simplest form.

What is the greatest possible product for a set of positive integers whose sum is 25?

What is the units digit of 10! in base 9?

For graduation the senior class is divided into 38 equal groups to be seated in an auditorium in which each row has 35 seats. Seniors fill all but the last seat in the last row. If there are less than 1,000 graduating seniors, how many students are in each of the 38 groups?

The tires on an antique car have wheels with five spokes, but the tires are slightly different sizes: The large tire in back has a diameter of 30 inches, while the smaller tire in front has a diameter of 27 inches. A photographer wants to photograph the car while the wheels are in a position identical to the one above (with both stars up-side down). How many times will this occur during the course of a quarter-mile parade if the wheels begin the parade in the position shown?

What is the smallest positive integer a for which both a and a + 1 each have exactly 6 factors?