Information about Arithmetic

SAT/GRE

Number systems What is the difference between number sense and counting? (Brahmagupta, 596 AD)

A history of number systems 46,206 = One important part of every number system is its base.

Types of numbers What types of numbers can you recall?

Types of numbers Term Definition Examples Natural numbers Counting numbers. 1, 2, 3, 4, 5, 6 Whole numbers Counting numbers and zero. 0, 1, 2, 3 Integers Whole numbers, their opposites and zero. –2, –1, 0, 1, 2, 3 Rational Numbers Repeating or terminal decimals. 1/3, 1/2 Irrational numbers Non-repeating and non terminating decimals 22/7 , Real numbers Every number on the number line Anything!

The set of numbers

Consecutive numbers and the number line

Odd and Even Numbers Addition Subtraction even + even = even even – even = even Multiplication even . even = even odd + odd = even odd – odd = even odd . odd = odd even + odd = odd even – odd = odd even . odd = even odd +even = odd odd – even = odd odd . even = even

Positives and Negatives Multiplication Division positive . positive = positive positive / positive = positive negative . negative = positive negative / negative = positive positive . negative = negative positive / negative = negative negative . positive = negative negative / positive = negative

Divisibility and Remainders What rules do you remember for divisibility? For example all even numbers are divisible by 2.

Divisibility and Remainders All whole numbers are divisible by 1. A number that ends in an even digit is divisible by 2. A number is divisible by 3 if its digits add up to a number divisible by 3. For example, 384 is divisible by 3 because 3 + 8 + 4 = 15, and 15 is divisible by 3. A number is divisible by 4 if its last two digits are divisible by 4. The number 5,764 is divisible by 4 because 64 is divisible by 4. A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 6 if it is even and divisible by 3. This rule is a combo of rules 2 and 3. Sadly, there is no rule for 7. A number is divisible by 8 if its last three digits are divisible by 8. For example, 1,249,216 is divisible by 8 because 216 is divisible by 8. A number is divisible by 9 if its digits add up to a number divisible by 9. The number 2,952 is divisible by 9 because 2 + 9 + 5 + 2 = 18. A number is divisible by 10 if it ends in 0.

Divisibility and Remainders: Squares

Divisibility and Remainders: Squares 0 -> then the ending digit is a 0 1 -> then the ending digit is 1 or 9. 4 -> then the ending digit is 2 or 8. 5 -> then the ending digit is a 5. 6 -> then the ending digit is 4 or 6. 9 -> then the ending digit is 3 or 7. C. After finding the last digit (or possibility between two digits) mentally chop off the last two digits and focus on the remaining digits.

Example 1: Q) The number n is a 2 digit integer. When n is divided by 5 it leaves a remainder of 4 and when n is divided by 9, it leaves a remainder of 7. What is the value of n?

Strategies 1a) Picking Numbers Step 1. Pick Simple Numbers and substitute for variables. Step 2. Try Them Out Try out all the answer choices using the numbers you picked, eliminating those that gave you a different result. Step 3. Try Different Values If more than one answer choice works, use different values and start again 1b) Recognition by prolonged and multiple use of Picking Numbers

1a) Picking Numbers

1b) Recognition Use different and easy Number Picks to eliminate any possible integer answers.

2) Back-solving Step 1: Estimate the answer. Step 2: Plug in an answer. Step 3: Keep plugging in till you find one that works. Step 4: Try out all answers. Q. If x is an integer and 2 is the remainder when 3x + 4 is divided by 5, then x could equal (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

3) Elimination

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