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What does the absolute value of a number refer to? |
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Definition
How far away from 0 that number is on the number line... |
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On the number line, if two numbers are opposites of each other, what can you say about them? |
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Definition
That they have the same absolute value, and 0 is halfway between them. |
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If x = -y, then, on a number line, mark x and y... |
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Definition
There are 2 possible scenarios here (we can't tell which variable is positive without more information):
1) <-- x -- 0 -- y -->
or
2) <-- y -- 0 -- x -->
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How do you write remainders in fractional notation? |
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Definition
x/N = Q + R/N
i.e. dividend/divisor = Quotient + remainder/divisor |
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if x has a remainder of 3 when divided by 7, and y has a remainder of 2 when divided by 7, what's the remainder when x + y is divided by 7? |
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Definition
sol'n 1: pick suitable numbers for x and y... e.g. 14 + 3 = 17 (x) and 7 + 2 = 9 (y). Adding together, we see that 17 + 9 = 26, which has a remainder of 5 when divided by 7.
sol'n 2: Algebraically, you could write x = 7z + 3 and y = 7c + 2 (note z and c are arbitrary integers). So x + y = 7z + 3 + 7c + 2 = 7z + 7c + 5 = 7(z + c) + 5, which equals a multiple of 7, plus 5. Thus, the remainder is 5. |
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Give 2 properties of the remainder of any number. |
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Definition
The remainder of any number MUST be non-negative, and smaller than the divisor. |
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When you divide an integer by 7, what could the remainder be?
What is the rule for this? |
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Definition
0,1,2,3,4,5 or 6.
Notice you cannot have a remainder equal to or larger than 7, and that you have exactly 7 possible remainders.
This can be generalised. When you divide an integer by a positive integer N, the possible remainders range from 0 to (N - 1). There are thus N possible remainders. |
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If a/b yields a remainder of 5, c/d yields a remainder of 8, and a,b,c,d are all integers, what's the smallest possible value for b + d? |
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Definition
Since the remainder must be smaller than the divisor, 5 must be smaller than b. Similarly, 8 must be smaller than d. therefore the smallest possible value for b + d = 6 + 9 = 15 |
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An integer x divided by another integer y yields a remainder of 0. What can we say about this? |
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Definition
That x is divisible by y. |
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Discuss 2 ways of determing the number of total factors of an integer... |
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Definition
1. Use Factor pairs. This is only useful for small numbers, however.
2. For larger numbers: If a number has prime factorisation ax x by x cz (where a,b & c are all prime) then the number has
(x + 1)(y + 1)(z + 1) factors.
Another way of saying this is: "The following formula can be used to find the number of divisors of any given number. Factor 90 ( [image] ) and then multiply the powers+1 [image]"
In addition, if a prime factor appears to the Nth power, then there are (N + 1) possibilities for the occurences of that prime factor. This is true for each of the individual primer factors of any number.
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Explain why the following 3 properties of the GCF and LCM are true:
1. (GCF of M & N) x (LCM of M & N) = M x N
2. GCF of M and N can't be larger than the difference between M and N
3. Consecutive multiples of N have a GCF of N |
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Definition
1. (GCF of M & N) x (LCM of M & N) = M x N
This is because the GCF is composed of the SHARED prime factors of M and N. The LCM is composed of all the other, or NON-SHARED prime factors of M and N. Thus, thinking about the VENN diagram way of looking at GCF and LCM, ... ? (FILL THIS IN...)
2. GCF of M and N can't be larger than the difference between M and N
3. Consecutive multiples of N have a GCF of N |
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| What is the "commutative property?" |
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Definition
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| What is the "distributive property?" |
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Definition
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| What is the "associative property?" |
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Definition
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This pattern holds true for the multiples of any integer N.
If you add or subtract multiples of N, the result is a multiple of N.
You can restate this principle using any of the disguises above: for instance, if N is a divisor of x and of y, then N is a divisor of x +y. |
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Definition
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| list all the different ways you can think of to say the same thing as '12 is divisible by 3' |
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Definition
• 12 is divisible by 3
• 3 is a divisor of 12, or 3 is a factor of 12
• 12 is a multiple of 3
• 3 divides 12
• 12/3 is an Integer
• 12/3 yields a remainder of 0
• 12 = 3n, where n is an integer
• 3 "goes into" 12 evenly
• 12 items can be shared among 3
people so that each person has
the same number of items. |
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List all the primes up to 100 |
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Definition
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 .. |
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| Explain the factor foundation rule |
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Definition
if a is a factor of b,and b is a factor of c, then a is a factor of c.
In other words, any integer is divisible by all of.its factors-and it is also divisible by all of the FACTORS of its factors.
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| what happens when you add a multiple of N to a non-multiple of N ?? |
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Definition
When you add a multiple of N to a non-multiple of N, the result is a non-multiple of N. (The same holds true for subtraction.)
18 - 10 = 8 (Multiple of 3) - (Non-multiple of 3) = (Non-multiple of 3) |
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What happens if you add two non-multiples of N ?
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Definition
If you add two non-multiples of N, the result could be either a multiple of N or a non-multiple of N.
19 + 13 = 32 (Non-multiple 00) + (Non-multiple 00) = (Non-multiple 00)
19 + 14 = 33 (Non-multiple of 3) + (Non-multiple of 3) = (Multiple of 3)
The exception to this rule is when N=2. Two odds always sum to an even. |
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