Factors

Factors

Suppose you were given any positive integer and wanted to determine how many positive, integer divisors it had

but were unwilling to write them all out?

How could you do this?

Red Question: How many positive, integer divisors does the number 630 have?

(Special note: Keep in mind that the question is asking you to determine how many divisors a number has, not what those divisors are! To appreciate the difference, consider a number like 2304. You can determine rather quickly that it has 27 divisors. Finding those divisors would take quite a while.)

A complete solution to this question is posted below.

Green Question: A positive integer has exactly eight divisors, including 1 and itself. If two of the divisors are

35 and 77, what is the sum of all eight divisors?

Blue Question: (PHTST vol.1) Given that and are perfect squares, where and are positive integers,

determine the minimum value of

Orange Question: Prove that

is irrational.

Yellow Question: If

and are distinct odd primes and , show that has at least 4 distinct prime divisors.

STEP 2009, Math I, #1: (no calculator) A proper factor of an integer N is a positive integer, not 1 or N, that divides N.

(i.) Show that has exactly 10 proper factors. Determine how many other integers of the form

(where

and are integers) have exactly 10 proper factors.

(ii.) Let N be the smallest positive integer that has exactly 426 proper factors. Determine N, giving your

answer in terms of its prime factors.

STEP 2005, Math II, #2: For any positive integer

, the function is defined by

where

are the only prime numbers that are factors of .

Thus

(a) (i) Evaluate and .

(ii) Show that is an integer for all .

(b) Prove, or disprove by means of a counterexample, each of the following:

(i) (ii)

if and are distinct prime numbers.

(iii)

only if and are distinct prime numbers.

and a prime number such that .

STEP 2006, Math I, #1: Find the integer

that satisfies . Find also a small integer

such that

is a perfect square. Hence express 33127 in the form , where and are integers greater than 1.

By considering the possible factorizations of 33127 , show that there are exactly two values of

for which

is a perfect square, and find the other value.

A frequent technique used for proving facts about numbers is an indirect proof. In this case, you assume the opposite of what you want

to prove and show that this leads to a contradiction. Here is a classic proof by introduction.

Mulitples and non-Multiples

When proving facts about integers, it can be helpful to know how to write some properties using algebraic notation.

is a multiple of then that means that there is another integer such that .

For example, if

is a multiple of the number 3, that means that there exists an integer such that .

is NOT a multiple of the number 3, then we can still represent algebraically since there are only two other possibilities: either is 1 more that

a multiple of 3 or

is 2 more than a multiple of 3. That means there are three possible representations for .

We can also use this technique to investigate properties of squares.

Example: Show that no perfect square integer has a units digit equal to 2, 3, 7, or 8.

Solution: If

is any integer, then

No other possibilities exist. We examine each case in turn.

then which means that is a multiple of 5.

then so is 1 more than a multiple of 5.

then so is 4 more than a multiple of 5.

then so is 1 more than a multiple of 5.

Thus, any perfect square is either a multiple of 5, 1 more than a multiple of 5, or 4 more than a multiple of 5. No perfect square is 2 or 3 more

than a multiple of 5.Hence, no perfect square could end in a 2 or 7, a 3 or an 8.

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