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If you subtract the remainder from the dividend, the resulting # is divisible by the divisor.
ex. r=1, Dividend (N) = 31, Divisor (D)= 3
N  r = # divisible by D
31  1 = 30 (divisible by 3)

If you add the difference b/w the divisor and the remainder to the dividend, the resulting # will be divisible by the divisor.
ex. r=1, Dividend (N) = 31, Divisor (D)= 3
Dr = # (dif b/w Divisor & remainder)
31 = 2
Dif + Dividend = # divisible by divisor
2 + 31 = 33 (divisible by divisor 3)

When a smaller integer is divided by a greater integer, the quotient is 0 and the remainder is the dividend.
ex. 8 ÷ 17 = 0 r8

When working with complex fractions, always start with the innermost fraction first.

If a # has exactly three factors...
it must be the square of a prime #
 ex. 9 = 3^{2}
 Factors of 9: 9, 3, 1

All positive integers other than 1 have an even # of factors, unless the # is a perfect square.
 ex: Factors of 12: 1, 2, 3, 4, 6, 12
 (6 factors)
 ex: Factors of 15: 1, 3, 5, 15
 (4 factors)
 ex: Factors of 16: 1, 2, 4, 8, 16
 (5 factors) 16 = perfect square of 4

Sometimes the fastest way to solve a ratio problem is to set up a proportion.
Ex:
3 = 9
4 _x

Remember to solve for any quantity on a ratio problem, you must have the multiplier or at least one amount.

If you are given one ratio and a certain # is added/subtracted to the ratio, the new ratio can't be determined unless the original amount is given.

All ratios can be expressed with a multiplier:
Ex: 3/5 = 3x/5x or 3:5 = 3x:5x

Percents: When increasing a value by a certain % and then decreasing it by the same %, the end result will always be less than the orig. value.
Ex: 10 + 20% = 12, 12  20% = 9.6

Setting an exponent to Zero is an easy way to create a number (1) that divides evenly into any integer.

If wording "percent greater" is used, what do you need to do?
Add the starting # to the new # (starting # * # was increased by).
 Ex: LY = $12mil, TY = 150 percent greater
 Then TY = ? (1.5 * 12) > (18) + 12 = 30

If wording is "percent of _____", the #/wording which comes after the "of" or "per" goes where?
Under the fraction line

Remember if you have a variable with no coefficient, i.e. coefficient = 1, remember you can turn into a fraction if needed to combine terms.
i.e. 1C = 5/5 C


