Properties of Numbers

Question 1

Testing Numbers

Answer 1

Test the following and be strategic.

• Positive integer
• Positive proper fraction
• 0
• Negative proper fraction
• Negative integer

Question 2

Systems of equations: Avoid the C trap for data sufficiency.

We can sometimes solve a single equation containing 2 variables.

Answer 2

If x and y are positive integers, what is the sum of x and y?

1) 5x + 9y = 63
2) xy = 18

x = 9(7 - y) / 5
→ The only way (7 - y) can be divisible by 5 is if y = 2.
→ Answer is A.

Question 3

Prime Numbers < 100

Answer 3

Trick: 1, 3, 5, 7, 9 units digit

Remember: 1 is NOT a prime number

There are 25 prime numbers between 1 and 100

Question 4

Integers with exactly 3 factors

Answer 4

The only integers with exactly 3 factors are the squares of prime numbers

e.g., 9 (1, 3, 9), 25 (1, 5, 25)

Question 5

Product of Any n Consecutive Integers

Answer 5

The product of any n consecutive integers must be divisible by all of the factors of n!

• e.g., 20 x 21 x 22 x 23 is divisible by 4!

Question 6

Product of n Consecutive Even Integers

Answer 6

The product of n consecutive even integers will always be divisible by 2ⁿ x n!

(There’s no special divisibility rule for the product of n consecutive odd integers)

Question 7

Product of 2 Consecutive Even Integers

Answer 7

Divisible by 8

Because 2² x 2! = 2³ = 8

Question 8

Whole Number Definition

Answer 8

Nonnegative integers (so includes 0)

Question 9

Even and Odd Number: Addition and Subtraction

Answer 9

Assuming 2 numbers in operations:

• Same types of integers in operations → Even
• Different types of integers in operations → Odd

***Addition and subtraction rules are the same

• If x + y is even, then x - y must also be even
• If x + y is odd, then x - y must also be odd

Question 10

Even and Odd Number: Multiplication Rule

Answer 10

• Any number x Even is always Even
• Product of odd numbers is always odd

Question 11

Even and Odd Number: 2x + 1

Answer 11

If x is a positive integer, then:

• 2x + 1 → Odd
• 2x - 1 → Odd

Question 12

Even and Odd Number: Division Rules

Answer 12

• Even ÷ Odd = Even
• Odd ÷ Odd = Odd
• Odd ÷ Even doesn’t yield an integer
• Even ÷ Even = either Odd or Even

Question 13

Representing even/odd numbers algebraically

Answer 13

If n is a positive integer:

• n is even: n = 2k, where k is a nonnegative integer
• n is odd: n = 2k + 1, where k is a nonnegative integer

Question 14

Properties of 0

Answer 14

• 0 is the only number whose opposite is itself
• 0 raised to the zero power is undefined
• 0 is a multiple of all numbers
• 0 is not a factor of any number except itself
• Any number (except for 0) raised to the power of 0 is 1

Question 15

Properties of 1

Answer 15

1 is not a prime number

Question 16

Reciprocals: 1 and -1

Answer 16

1 and -1 are the only numbers whose reciprocals are equal to themselves

Question 17

Trailing zeros are created by (5 x 2) pairs

Answer 17

The number of trailing zeros of a number is the number of (5 x 2) pairs in the prime factorization of that number

Question 18

Remainders after division by 10ⁿ

Answer 18

• When a whole number is divided by 10, the remainder will be the units digit of the dividend.
• When a whole number is divided by 100, the remainder will be the last two digits of the dividend, etc.

Question 19

Remainders after division by 5

Answer 19

When integers with the same units digit are divided by 5, the remainder is constant.

• 7/5 = 1…2
• 17/5 = 3…2
• 57/5 = 11…2

Question 20

Remainder Patterns

Answer 20

What is the least possible value that can be subtracted from 2⁵⁸⁶ so that the result is a multiple of 7?

• Write out the initial patterns of remainders: 2⁰ / 7, 2¹ / 7 ….etc.
• Remainder pattern is 2-4-1
• 585 is the closest number to 586 that is divisible by 3
• So 586 means the remainder is 2
• Answer = subtract out 2

Question 21

If √x is an integer

Answer 21

that means x is a non-negative integer