Divisibility Tests

We use the notation a/b to say “a divides b”. For example, 2/10 would tell us that “2 divides 10”.

Divisibility by two: If the last digit is even, the original number is divisible by 2.
Example that works 314
Example that doesn’t work 413

Divisibility by three: If the sum of the digits is divisible by 3, then the original number is divisible by 3.
Example that works 315
Example that doesn’t work 314

Divisibility by four: If the last two digits form a number that is divisible by 4, then the original number is divisible by 5.
Example that works 316
Example that doesn’t work 315

Divisibility by five:  If the last digit is either a 0 or 5, then the original number is divisible by 5.
Example that works 320
Example that doesn’t work 513

Divisibility by six:  If the number is divisible by 2 and 3, then it is divisible by 6.
Example that works 414
Example that doesn’t work 314

Divisibility by seven: Doesn’t work. Just preform the division.

Divisibility by eight: if the last three digits form a number that is divisible by 8, then the original number is divisible by 9.
Example that works 1096
Example that doesn’t work 403316

Divisibility by nine: If the sum of the digits is divisible by 9, then the original number is divisible by 9.
Example that works 315
Example that doesn’t work 317

Divisibility by ten: If the last digit is 0, then the original number is divisible by ten.
Example that works 520
Example that doesn’t work 523

Divisibility by eleven: Find the sum of the odd-numbered digits (odd sum) and the sum of the even numbered digits (even sum). Take the difference between odd sum and even sum. If this difference is divisible by 11, then the original number is divisible by 11.

Divisibility by twelve: If the number is divisible by 3 and 4, then it is divisible by 12.
Example that works 1308
Example that doesn’t 1433

 

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Greatest Common Factor

For any two numbers, there is always a number
that is a factor of both. For example, the numbers
24 and 36 both have 6 as a factor. When a number
is a factor of two numbers, it is called a common
factor
or common divisor. 
 
Among the common factors of two numbers, there
will always be a largest number, which is called the
greatest common factor. The greatest common factor
of 24 and 36 is 12. This is sometimes written as GCF(24,36)=12.
               For any two nonzero whole numbers a and b the
greatest common factor, written as GCF(a,b), is the greatest
factor (divisor) of both a and b.

Least Common Multiple

Every pair of nonzero whole numbers has an infinite number of
common multiples. Among these common multiples will always
be a smallest number, which is called the least common multiple.
For example, the least common multiple of 5 and 7 is 35. This is
sometimes written as LCM(5,7) = 35. For any two nonzero whole
numbers a and b, the least common multiple, written LCM(a,b),
is the smallest multiple of both a and b.

Prime Factorizations

Composite number can always be written as a product
of primes. Such a product is called the Prime factorization
of a number. For example, the prime factorization of 12 is
2 x 2 x 3. Since every whole number greater than 1 is either
prime or a product of primes, prime numbers are often
referred to as the building blocks of the whole numbers.

Babylonian Number System

The Babylonians developed a base-sixty numeration
system. Their basic symbols for 1 through 59 were
additively formed by repeating different symbols.
To write numbers greater than 59, the Babylonians
used their basic symbols for 1-59 and the concept of
Place Value. Place value is a power of the base, and the
Babylonian place values were 1, 60, 602, 603, etc. Their
basic symbols had different values depending on the
position or location of the symbol.

For practice with this number system click here