Playing with Multiples of N:


(I)
If you add/subtract multiples of number 'N', the result is also a multiple of 'N'.

Examples:
35+21 = 56 [Multiple of 7]
20-15 = 5 [Multiple of 5]


TAKEAWAY:
In general, if N is a divisor of both x and y, then N is a divisor of both x+y and x-y.


(II)
If you add/subtract a multiple of N to/from a non-multiple of N, the result is a non-multiple of N.

Example:
9-5 = 4 [(Multiple of 3) - (Non-Multiple of 3) = (Non-multiple of 3)]



(III)
If you add/subtract 2 non-multiples of N, the result could either be a multiple or a non-multiple of N.

Examples:
19+13 = 32 [(Non-Multiple of 3) - (Non-Multiple of 3) = (Non-multiple of 3)]
19+14 = 33 [(Non-Multiple of 3) - (Non-Multiple of 3) = (Multiple of 3)]

EXCEPTION:
When N = 2, two odds always sum to an even number (Multiple of 2).

GCF Facts:


1. GCF of integers 'm' and 'n' CANNOT be larger than the difference between 'm' and 'n'.

Assume that GCF of m and n is 12. m and n are both multiples of 12. Consecutive multiples of 12 are 12 units apart from each other on the number line. Therefore, m and n CANNOT be less than 12 units apart.


2. Consecutive multiples of n have a GCF of n.

4 and 8 are multiples of 4. Thus 4 is a common factor of both the numbers. 4 and 8 are exactly 4 units apart from each other on the number line. Thus, 4 is the greatest common factor (GCF) of 4 and 8. That is why GCF of any 2 consecutive numbers is ALWAYS 1 as both are multiples of 1.

Consecutive Integers:


1. In any set of 3 consecutive integers, one of the integers is ALWAYS divisible by 3.



2. (x-1) * x * (x+1) is divisible by 8, if x is odd.

(x-1) and (x+1) are even. (x-1) is atleast divisible by 2 (Since it always has a 2). As (x-1) and (x+1) are consecutive multiples of 2, (x+1) will have an additional 2 units apart from another 2 (i.e. 4). So, (x+1) is divisible by 4.
Thus, (x-1) * x * (x+1) is divisible by 8

Factor Foundation Rule:

"If x is a factor of y and y is a factor of z, then x is a factor of z".

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.

Example:
Consider 36.

36 can be broken down into 12 * 3

12 can be broken down into 2^2 * 3

So, 2^2 is also a factor of 36

Divisibility Rules - ODDs and EVENs:

In general,

Odd integer divided by any other integer CANNOT produce an even integer.
Odd integer divided by an even integer CANNOT produce and integer.

Exponent Rules:

1. x^a * x^b = x^(a+b)

2. (a^x)^y = a^xy = (a^y)^x

3. a^x * b^x = (ab)^x

4. x^(a/b) = b root (x^a) = (b root (x))^a

5. x^a / x^b = x^(a-b)

6. (a/b)^x = a^x / b^x

7. x^-a = 1 / x^a

8. a^x + a^x + a^x = 3 * a^x

Properties of Roots:

1. n root(x) / n root(y) = n root(x/y)

2. n root(x) * n root(y) = n * root(xy)

3. b root(x^a) = (b root(x)) ^ a = x^(a/b)

Evenly Spaced Sets:

i. Mean and Median are equal
ii. Mean and Median of the set = Average (First + Last terms)
iii. Sum of the elements = Mean of the set * Number of elements in the set

Counting Integers:

For Consecutive integers:

(Last term - First term + 1)


For Consecutive multiples:

[(Last term - First term) / increment] + 1, where increment is the difference between each consecutive term in the set.

Average of an ODD number of consecutive integers will ALWAYS be an integer.

E.g: Average of 5 numbers (1,2,3,4,5) is 3 (an integer).


Average of an EVEN number of consecutive integers will NEVER be an integer.

E.g: Average of 5 numbers (1,2,3,4,5,6) is 3.5 (NOT an integer).

For any set of consecutive integers with an ODD number of items, the sum of all integers is ALWAYS a multiple of the number of items.

Example:
Consider 1,2,3,4,5. Number of items is 5.

1+2+3+4+5 = 15. 15 is a multiple of 5 (the number of items).



For any set of consecutive integers with an EVEN number of items, the sum of all integers is NEVER a multiple of the number of items.

Example:
Consider 1,2,3,4,5,6. Number of items is 6.

1+2+3+4+5+6 = 21. 21 is NOT a multiple of 6 (the number of items).


--[You can try out any set of numbers to see whether the rule holds]--

Terminating/Non-Terminating decimals: (VERY IMPORTANT)

If the fraction is completely simplified and the denominator only have 2's and 5's (or) only 2's or only 5's, then it is a terminating decimal.


If the fraction is completely simplified and the denominator of the fraction has prime factors other than 2's and/or 5's , then it is a repeating/non-terminating decimal.