Number System and Divisibility

Concept

The sets of numbers

Numbers stack in layers, each containing the one before it.

Natural numbers: 1, 2, 3, ... — the counting numbers.
Whole numbers: the naturals together with 0.
Integers: the whole numbers together with their negatives: ..., -2, -1, 0, 1, 2, ...
Rational numbers: anything expressible as a ratio p/q of two integers with q not zero. Their decimal form either terminates (0.75) or repeats (0.333...).
Irrational numbers: real numbers that are not rational — their decimals neither terminate nor repeat, such as the square root of 2 or pi.
Real numbers: the rationals and irrationals together.

Among the naturals, a prime has exactly two factors, 1 and itself (2, 3, 5, 7, 11, ...); a composite has more than two. 1 is neither. 2 is the only even prime.

Method

Divisibility rules

A number is divisible by:

2 — its last digit is even.
3 — the sum of its digits is divisible by 3.
4 — the number formed by its last two digits is divisible by 4.
5 — its last digit is 0 or 5.
6 — it passes the tests for both 2 and 3.
8 — the number formed by its last three digits is divisible by 8.
9 — the sum of its digits is divisible by 9.
10 — its last digit is 0.
11 — the difference between the sum of the digits in odd positions and the sum in even positions is 0 or a multiple of 11.

For a composite divisor made of coprime parts, test the parts: divisibility by 12 means divisibility by 3 and by 4; by 72 means by 8 and by 9.

Method

Counting factors from a prime factorisation

Write the number as a product of prime powers: N equals p1 to the power a, times p2 to the power b, times p3 to the power c, and so on.

Number of factors equals (a plus 1) times (b plus 1) times (c plus 1) ...
Number of factors that are perfect squares counts only the even exponents 0, 2, 4, ... in each bracket.
Sum of all factors equals the product, over each prime, of (p to the power (exponent plus 1) minus 1) divided by (p minus 1).

A number has an odd number of factors exactly when it is a perfect square, because only then does one factor (the square root) pair with itself.

Concept

Unit digit of a power

The unit digit of successive powers of a fixed digit repeats in a short cycle.

2: 2, 4, 8, 6 (length 4)
3: 3, 9, 7, 1 (length 4)
7: 7, 9, 3, 1 (length 4)
8: 8, 4, 2, 6 (length 4)
4: 4, 6 (length 2); 9: 9, 1 (length 2)
0, 1, 5, 6: always 0, 1, 5, 6 (length 1)

To find the unit digit of a base raised to a power, take the unit digit of the base, find the remainder when the exponent is divided by the cycle length, and read that position in the cycle. A remainder of 0 means the last entry of the cycle.

Worked example

Worked example: divisibility checks

Is 453,192 divisible by 8? By 9?

By 8: look at the last three digits, 192. Since 192 divided by 8 is 24, a whole number, 453,192 is divisible by 8.

By 9: the digit sum is 4 plus 5 plus 3 plus 1 plus 9 plus 2, which is 24. Since 24 is not divisible by 9, 453,192 is not divisible by 9. (It is divisible by 3, because 24 is.)

Worked example

Worked example: counting factors

How many factors does 360 have, and how many of them are odd?

Prime factorise: 360 equals 2 to the power 3, times 3 to the power 2, times 5 to the power 1.

Number of factors: (3 plus 1) times (2 plus 1) times (1 plus 1), which is 4 times 3 times 2, equal to 24.

Odd factors: ignore the power of 2 (an odd factor uses 2 to the power 0). That leaves (2 plus 1) times (1 plus 1), which is 3 times 2, equal to 6.

Worked example

Worked example: unit digit of a power

Find the unit digit of 7 raised to the power 83.

The unit digit of 7's powers cycles as 7, 9, 3, 1, with length 4.

Divide the exponent by 4: 83 divided by 4 leaves a remainder of 3.

The 3rd entry in the cycle 7, 9, 3, 1 is 3. So the unit digit of 7 to the power 83 is 3.

As a check, 7 to the power 4 ends in 1, so 7 to the power 80 ends in 1, and 7 to the power 83 ends like 7 cubed, which is 343 — unit digit 3.

Common mistake

Common mistakes

Calling 1 a prime. A prime has exactly two distinct factors; 1 has one.
Testing divisibility by 8 with two digits. Use the last three digits for 8, two for 4.
Forgetting to reduce the exponent modulo the cycle length for unit digits. A remainder of 0 points to the last entry of the cycle, not the first.
Adding 1 to the wrong number when counting factors. It is (exponent plus 1) for each prime, then multiply the brackets.
Assuming an even number of factors. Perfect squares have an odd number.
Mixing up the rule for 11. It is the alternating sum (odd positions minus even positions), not the plain digit sum.

Quick revision

Sets: natural, whole, integer, rational (terminating or repeating decimals), irrational, real. Primes have exactly two factors; 1 is neither; 2 is the only even prime.
Divisibility: 2 (even last digit), 3 (digit sum), 4 (last two digits), 5 (0 or 5), 6 (2 and 3), 8 (last three digits), 9 (digit sum), 11 (alternating digit sum). For composite divisors, test coprime parts.
Factor count from N = p1^a p2^b ... is (a+1)(b+1)... ; odd number of factors means a perfect square.
Unit digit of a power: take the base's unit digit, reduce the exponent modulo the cycle length (usually 4), read that position; remainder 0 means the last entry.

Practice

Is 7,20,918 divisible by 11?
How many factors does 1,200 have?
Find the unit digit of 3 raised to the power 100.
What is the smallest number that must be added to 5,834 to make it divisible by 9?
How many numbers between 1 and 200 are divisible by 6 but not by 4?
The number 34A6 is divisible by 4. What digit can A be?
What is the largest three-digit number that is exactly divisible by 17?
Find the remainder when the sum 1 + 2 + 3 + ... + 100 is divided by 7.