Essential Number Properties and Divisibility Rules

Fundamental Number Concepts

Prime Numbers Up to 97

Here is a list of prime numbers up to 97:

• 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

Rules for Exponents

Understanding how exponents work is crucial for various mathematical operations:

• Positive Base: If the base (m) is positive, the result (b) of mn = b is always positive.
• Negative Base, Even Exponent: If the base (-M) is negative and the exponent (n) is an even number, the result (+b) of (-M)n = +b is positive.
• Negative Base, Odd Exponent: If the base (-M) is negative and the exponent (n) is an odd number, the result (-b) of (-M)n = -b is negative.

Powers of Decimal Numbers

Here are examples of squaring and cubing decimal numbers:

• 1.12 = 1.21
• 0.42 = 0.16
• 0.032 = 0.0009
• 0.23 = 0.008

Divisibility Criteria

Divisibility rules help determine if a number can be evenly divided by another without performing the actual division.

Criteria for Divisibility by 2

A number is divisible by 2 if it ends in zero or an even number.

Examples: 24, 238, 1024.

Criteria for Divisibility by 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Examples:

• 564 (5 + 6 + 4 = 15, which is a multiple of 3)
• 2040 (2 + 0 + 4 + 0 = 6, which is a multiple of 3)

Criteria for Divisibility by 5

A number is divisible by 5 if it ends in zero or five.

Examples: 45, 515, 7525.

Criteria for Divisibility by 7

A number is divisible by 7 if the difference between the number formed by its digits (excluding the units digit) and twice the units digit is 0 or a multiple of 7.

Examples:

• 343 (34 - 2 × 3 = 28, which is a multiple of 7)
• 105 (10 - 5 × 2 = 0)
• 2261 (226 - 1 × 2 = 224. Repeat the process with 224: 22 - 4 × 2 = 14, which is a multiple of 7.)

Criteria for Divisibility by 11

A number is divisible by 11 if the difference between the sum of its digits in odd places and the sum of its digits in even places is 0 or a multiple of 11.

Examples:

• 121 ((1 + 1) - 2 = 0)
• 4224 ((4 + 2) - (2 + 4) = 0)

Additional Divisibility Criteria

Criteria for Divisibility by 4

A number is divisible by 4 if its last two digits are zero or a multiple of 4.

Examples: 36, 400, 1028.

Criteria for Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Examples: 72, 324, 1506.

Criteria for Divisibility by 8

A number is divisible by 8 if its last three digits are zero or a multiple of 8.

Examples: 4000, 1048, 1512.

Criteria for Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Examples:

• 81 (8 + 1 = 9)
• 3663 (3 + 6 + 6 + 3 = 18, which is a multiple of 9)

Criteria for Divisibility by 10

A number is divisible by 10 if its units digit is 0.

Examples: 130, 1440, 10230.

Criteria for Divisibility by 25

A number is divisible by 25 if its last two digits are zero or a multiple of 25.

Examples: 500, 1025, 1875.

Criteria for Divisibility by 125

A number is divisible by 125 if its last three digits are zero or a multiple of 125.

Examples: 1000, 1125, 4250.