Number properties
Divisibility & Primes
Arithmetic rules
The sum, difference and product of two integers is always an integer.
The result of dividing two integers is sometimes an integer. An integer is divisible by another
number if the integer can be divided by that number with an integer result (there is no
remainder).
8/2 = 4 we can say that 2 is a divisor or factor of 8.
Rules of divisibility by certain integers
This is an important shortcut to determine whether an integer is divisible by 2,3,4,5,6,8,9 and
10. An integer is divisible by:
2. If the integer is even. With a large number you can check the last numbers. If the digit is
0,2,4,6 or 8 than it is divisible by 2.
3. If the SUM of the integer’s DIGITS is divisible by 3. 72 is divisible by 3, because 7+3 = 9.
4. If the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4. 28/2 =
14/2 = 7. For large number only check the last two digits.
5. If the integer ends in 0 or 5. VB 75 and 80
6. If the integer is divisible by BOTH 2 and 3.
7. ..
8. If the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible
by 8. 32/2=16/2=8/2=4/2=2.
9. If the SUM of the integer’s DIGITS is divisible by 9. 4185 is divisible by 9, because
4+1+8+5 = 18
10. If the integer ends in 0.
Factors and multiples
A factor is a positive integer that divides evenly into an integer. 1,2,4 and 8 are all the
factor of 8.
A multiple of an integer is formed by multiplying that integer by any integer, so 8,16,24 are
some of the multiples of 8.
It is easy to confuse factors and multiples. The mnemonic: Fewer factors, more multiples
should help to remember the difference. An integer has a limited of factors, but there is an
infinite number of multiples.
If you add or subtract multiples of N, the result is a multiple of N. If N is a divisor of X and of
y, then N is a divisor of x+y.
Prime factorization
This tool can help you to determine all factors of that number. If the problem states or
assumes that a number is an integer, you may need to use prime factorization to solve the
problem.
The FACTOR FOUNDATION RULE is: if A is a factor of B, and B is a factor of c, then A is a
factor of C. Any integer is divisible by all of its factors and it is also divisible by all of the
factors of its factors.
The primebox
A prime box is a box that holds all the primes factors of a number. You can use the box to
test whether or not a specific number is a factor of another number. P19
, Remainders
Every division has 4 parts:
1. The DIVIDEND is het number begin dived.
2. The DIVISOR is the number hat is dividing.
3. The QUOTIENT is the number of times that de divisor goes into the dividend
completely.
4. The REMAINDER is what is left over if the dividend is not divisible by the divisor.
Three ways to express remainders
VB 17 = 3x5 +2
1. 17/5 = 3 + 2/5
Dividend/divisor = quotient + remainder/divisor
2. 17/5 = 3,4
3. 17/5 = 3 with a remainder of 2
Creating numbers with a certain remainder
When positive integer N is divided by 7, there is a remainder of 2. What are the possible
values of N. Try the integer remainder relationship:
Dividend = quotient x divisor + remainder.
N = ? x 7 +x
Integer x 7 + 2 = N
(quotient) (divisor) (reaminder) (dividend)
0 X 7 + 2 = 2
1 X 7 + 2 = 9
2 X 7 + 2 = 16
3 x 7 + 2 = 23
Odds, evens, positives & negatives
Arithmetic rules of odds & evens
Odd +/- Even = ODD
Odd +/- Odd = EVEN
Even +/- Even = Even
Odd x Odd = ODD
Even x Even = Even (and divisible by 4)
Odd x Even = Even
There are no guaranteed outcomes in division, but an odd number divided by any other
integer CANNOT produce an even integer. Also, an odd number divided by an even number
CANNOT produce an integer.
The sum of two primes
The sum of two primes is always even (odd + odd = even). If the answer is an oneven
number you know that one of the primes is 2 (the only even prime number).
Representing evens and odd algebraically
Even numbers are multiples of 2, so an arbitrary even number can be written as 2N (N is an
integer). Odd number are one more or less than multiples of 2, so can be written as 2N + 1 of
2N -1.
Divisibility & Primes
Arithmetic rules
The sum, difference and product of two integers is always an integer.
The result of dividing two integers is sometimes an integer. An integer is divisible by another
number if the integer can be divided by that number with an integer result (there is no
remainder).
8/2 = 4 we can say that 2 is a divisor or factor of 8.
Rules of divisibility by certain integers
This is an important shortcut to determine whether an integer is divisible by 2,3,4,5,6,8,9 and
10. An integer is divisible by:
2. If the integer is even. With a large number you can check the last numbers. If the digit is
0,2,4,6 or 8 than it is divisible by 2.
3. If the SUM of the integer’s DIGITS is divisible by 3. 72 is divisible by 3, because 7+3 = 9.
4. If the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4. 28/2 =
14/2 = 7. For large number only check the last two digits.
5. If the integer ends in 0 or 5. VB 75 and 80
6. If the integer is divisible by BOTH 2 and 3.
7. ..
8. If the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible
by 8. 32/2=16/2=8/2=4/2=2.
9. If the SUM of the integer’s DIGITS is divisible by 9. 4185 is divisible by 9, because
4+1+8+5 = 18
10. If the integer ends in 0.
Factors and multiples
A factor is a positive integer that divides evenly into an integer. 1,2,4 and 8 are all the
factor of 8.
A multiple of an integer is formed by multiplying that integer by any integer, so 8,16,24 are
some of the multiples of 8.
It is easy to confuse factors and multiples. The mnemonic: Fewer factors, more multiples
should help to remember the difference. An integer has a limited of factors, but there is an
infinite number of multiples.
If you add or subtract multiples of N, the result is a multiple of N. If N is a divisor of X and of
y, then N is a divisor of x+y.
Prime factorization
This tool can help you to determine all factors of that number. If the problem states or
assumes that a number is an integer, you may need to use prime factorization to solve the
problem.
The FACTOR FOUNDATION RULE is: if A is a factor of B, and B is a factor of c, then A is a
factor of C. Any integer is divisible by all of its factors and it is also divisible by all of the
factors of its factors.
The primebox
A prime box is a box that holds all the primes factors of a number. You can use the box to
test whether or not a specific number is a factor of another number. P19
, Remainders
Every division has 4 parts:
1. The DIVIDEND is het number begin dived.
2. The DIVISOR is the number hat is dividing.
3. The QUOTIENT is the number of times that de divisor goes into the dividend
completely.
4. The REMAINDER is what is left over if the dividend is not divisible by the divisor.
Three ways to express remainders
VB 17 = 3x5 +2
1. 17/5 = 3 + 2/5
Dividend/divisor = quotient + remainder/divisor
2. 17/5 = 3,4
3. 17/5 = 3 with a remainder of 2
Creating numbers with a certain remainder
When positive integer N is divided by 7, there is a remainder of 2. What are the possible
values of N. Try the integer remainder relationship:
Dividend = quotient x divisor + remainder.
N = ? x 7 +x
Integer x 7 + 2 = N
(quotient) (divisor) (reaminder) (dividend)
0 X 7 + 2 = 2
1 X 7 + 2 = 9
2 X 7 + 2 = 16
3 x 7 + 2 = 23
Odds, evens, positives & negatives
Arithmetic rules of odds & evens
Odd +/- Even = ODD
Odd +/- Odd = EVEN
Even +/- Even = Even
Odd x Odd = ODD
Even x Even = Even (and divisible by 4)
Odd x Even = Even
There are no guaranteed outcomes in division, but an odd number divided by any other
integer CANNOT produce an even integer. Also, an odd number divided by an even number
CANNOT produce an integer.
The sum of two primes
The sum of two primes is always even (odd + odd = even). If the answer is an oneven
number you know that one of the primes is 2 (the only even prime number).
Representing evens and odd algebraically
Even numbers are multiples of 2, so an arbitrary even number can be written as 2N (N is an
integer). Odd number are one more or less than multiples of 2, so can be written as 2N + 1 of
2N -1.
