# The Math League

## Whole Numbers

Whole Numbers and Their Basic Properties

Whole Numbers

The whole numbers are the counting numbers and 0. The whole numbers are 0, 1, 2, 3, 4, 5, ...

Place Value

The position, or place, of a digit in a number written in standard form determines the actual value the digit represents. This table shows the place value for various positions:

 Place (underlined) Name of Position 1 000 Ones (units) position 1 000 Tens 1 000 Hundreds 1 000 Thousands 1 000 000 Ten thousands 1 000 000 Hundred Thousands 1 000 000 Millions 1 000 000 000 Ten Millions 1 000 000 000 Hundred millions 1 000 000 000 Billions

Example:

The number 721040 has a 7 in the hundred thousands place, a 2 in the ten thousands place, a one in the thousands place, a 4 in the tens place, and a 0 in both the hundreds and ones place.

Expanded Form

The expanded form of a number is the sum of its various place values.

Example:

9836 = 9000 + 800 + 30 + 6.

Ordering

Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals.) For example, since 2 is smaller than 4 and 4 is larger than 2, we can write: 2 < 4, which says the same as 4 > 2 and of course, 4 = 4.

To compare two whole numbers, first put them in standard form. The one with more digits is greater than the other. If they have the same number of digits, compare the most significant digits (the leftmost digit of each number). The one having the larger significant digit is greater than the other. If the most significant digits are the same, compare the next pair of digits from the left. Repeat this until the pair of digits is different. The number with the larger digit is greater than the other.

Example: 402 has more digits than 42, so 402 > 42.

Example: 402 and 412 have the same number of digits. We compare the leftmost digit of each number: 4 in each case. Moving to the right, we compare the next two numbers: 0 and 1. Since 0 < 1, 402 < 412.

Rounding Whole Numbers

To round to the nearest ten means to find the closest number having all zeros to the right of the tens place. Note: when the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.

Examples:

Rounding 119 to the nearest ten gives 120.

Rounding 155 to the nearest ten gives 160.

Rounding 102 to the nearest ten gives 100.

Similarly, to round a number to any place value, we find the number with zeros in all of the places to the right of the place value being rounded to that is closest in value to the original number.

Examples:

Rounding 180 to the nearest hundred gives 200.

Rounding 150090 to the nearest hundred thousand gives 200000.

Rounding 1234 to the nearest thousand gives 1000.

Rounding is useful in making estimates of sums, differences, etc.

Example:

To estimate the sum 119360 + 500 to the nearest thousand, first round each number in the sum, resulting in a new sum of 119000 + 1000.. Then add to get the estimate of 120000.

Divisibility Tests

There are many quick ways of telling whether or not a whole number is divisible by certain basic whole numbers. These can be useful tricks, especially for large numbers.

Commutative Property of Addition and Multiplication

Addition and Multiplication are commutative: switching the order of two numbers being added or multiplied does not change the result.

Examples:

100 + 8 = 8 + 100

100 × 8 = 8 × 100

Associative Property

Addition and multiplication are associative: the order that numbers are grouped in addition and multiplication does not affect the result.

Examples:

(2 + 10) + 6 = 2 + (10 + 6) = 18

2 × (10 × 6) = (2 × 10) × 6 =120

Distributive Property

The distributive property of multiplication over addition: multiplication may be distributed over addition.

Examples:

10 × (50 + 3) = (10 × 50) + (10 × 3)

3 × (12+99) = (3 × 12) + (3 × 99)

Adding 0 to a number leaves it unchanged. We call 0 the additive identity.

Example:

88 + 0 = 88

The Zero Property of Multiplication

Multiplying any number by 0 gives 0.

Example:

88 × 0 = 0

0 × 1003 = 0

The Multiplicative Identity

We call 1 the multiplicative identity. Multiplying any number by 1 leaves the number unchanged.

Example:

88 × 1 = 88

Order of Operations

The order of operations for complicated calculations is as follows:

1) Perform operations within parentheses.

2) Multiply and divide, whichever comes first, from left to right.

3) Add and subtract, whichever comes first, from left to right.

Example:

1 + 20 × (6 + 2) ÷ 2 =

1 + 20 × 8 ÷ 2 =

1 + 160 ÷ 2 =

1 + 80 =

81.

Divisibility by 2

A whole number is divisible by 2 if the digit in its units position is even, (either 0, 2, 4, 6, or 8).

Examples:

The number 84 is divisible by 2 since the digit in the units position is 4, which is even.

The number 333336 is divisible by 2 since the digit in the units position is 6, which is even.

The number 1297000 is divisible by 2 since the digit in the units position is 0, which is even.

Divisibility by 3

A whole number is divisible by 3 if the sum of all its digits is divisible by 3.

Examples:

The number 177 is divisible by three, since the sum of its digits is 15, which is divisible by 3.

The number 8882151 is divisible by three, since the sum of its digits is 33, which is divisible by 3.

The number 162345 is divisible by three, since the sum of its digits is 21, which is divisible by 3.

If a number is not divisible by 3, the remainder when it is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.

Examples:

The number 3248 is not divisible by 3, since the sum of its digits is 17, which is not divisible by 3. When 3248 is divided by 3, the remainder is 2, since when 17, the sum of its digits, is divided by three, the remainder is 2.

The number 172345 is not divisible by 3, since the sum of its digits is 22, which is not divisible by 3. When 172345 is divided by 3, the remainder is 1, since when 22, the sum of its digits, is divided by three, the remainder is 1.

Divisibility by 4

A whole number is divisible by 4 if the number formed by the last two digits is divisible by 4.

Examples:

The number 3124 is divisible by 4 since the number formed by its last two digits, 24, is divisible by 4.

The number 1333336 is divisible by 4 since the number formed by its last two digits, 36, is divisible by 4.

The number 1297000 is divisible by 4 since the number formed by its last two digits, 0, is divisible by 4.

If a number is not divisible by 4, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.

Example:

The number 172345 is not divisible by 4, since the number formed by its last two digits, 45, is not divisible by 4. When 172345 is divided by 4, the remainder is 1, since when 45 is divided by 4, the remainder is 1.

Divisibility by 5

A whole number is divisible by 5 if the digit in its units position is 0 or 5.

Examples:

The number 95 is divisible by 5 since the last digit is 5.

The number 343370 is divisible by 5 since the last digit is 0.

The number 129700195 is divisible by 5 since the last digit is 5.

If a number is not divisible by 5, the remainder when it is divided by 5 is the same as the remainder when the last digit is divided by 5.

Example:

The number 145632 is not divisible by 5, since the last digit is 2. When 145632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2.

The number 7332899 is not divisible by 5, since the last digit is 9. When 7332899 is divided by 5, the remainder is 4, since 9 divided by 5 is 1 with a remainder of 4.

Divisibility by 6

A number is divisible by 6 if it is divisible by 2 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 6: its units digit is even and the sum of its digits is divisible by 3.

Examples:

The number 714558 is divisible by 6, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.

The number 297663 is not divisible by 6, since its units digit is not even.

The number 367942 is not divisible by 6, since it is not divisible by 3. The sum of its digits is 31, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 8

A whole number is divisible by 8 if the number formed by the last three digits is divisible by 8.

Examples:

The number 88863024 is divisible by 8 since the number formed by its last three digits, 24, is divisible by 8.

The number 17723000 is divisible by 8 since the number formed by its last three digits, 0, is divisible by 8.

The number 339122483984 is divisible by 8 since the number formed by its last three digits, 984, is divisible by 8.

If a number is not divisible by 8, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.

Example:

The number 172045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172345 is divided by 8, the remainder is 5, since when 45 is divided by 8, the remainder is 5.

Divisibility by 9

A whole number is divisible by 9 if the sum of all its digits is divisible by 9.

Examples:

The number 1737 is divisible by nine, since the sum of its digits is 18, which is divisible by 9.

The number 8882451 is divisible by nine, since the sum of its digits is 36, which is divisible by 9.

The number 762345 is divisible by nine, since the sum of its digits is 27, which is divisible by 9.

If a number is not divisible by 9, the remainder when it is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.

Examples:

The number 3248 is not divisible by 9, since the sum of its digits is 17, which is not divisible by 9. When 3248 is divided by 9, the remainder is 8, since when 17, the sum of its digits, is divided by 9, the remainder is 8.

The number 172345 is not divisible by 9, since the sum of its digits is 22, which is not divisible by 9. When 172345 is divided by 9, the remainder is 4, since when 22, the sum of its digits, is divided by 9, the remainder is 4.

Divisibility by 10

A whole number is divisible by 10 if the digit in its units position is 0.

Examples:

The number 1229570 is divisible by 10 since the last digit is 0.

The number 676767000 is divisible by 10 since the last digit is 0.

The number 129700190 is divisible by 10 since the last digit is 0.

If a number is not divisible by 10, the remainder when it is divided by 10 is the same as the units digit.

Examples:

The number 145632 is not divisible by 10, since the last digit is 2. When 145632 is divided by 10, the remainder is 2, since the units digit is 2.

The number 7332899 is not divisible by 10, since the last digit is 9. When 7332899 is divided by 10, the remainder is 4, since the units digit is 9.

Divisibility by 11

Starting with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11.

Examples:

Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.

Is the number 49137 divisible by 11? Starting with the units digit, add every other number:7 + 1 + 4 = 12. Then add the remaining numbers: 3 + 9 = 12. Since the difference between these two sums is 0, which is divisible by 11, 49137 is divisible by 11.

Is the number 16370706 divisible by 11? Starting with the units digit, add every other number:6 + 7 + 7 + 6 = 26. Then add the remaining numbers: 0 + 0 + 3 + 1=4. Since the difference between these two sums is 22, which is divisible by 11, 16370706 is divisible by 11.

Divisibility by 12

A number is divisible by 12 if it is divisible by 4 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 12: its last two digits are divisible by 4 and the sum of its digits is divisible by 3.

Examples:

The number 724560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is divisible by 3.

The number 36297414 is not divisible by 12, since the number formed by its last two digits, 14, is not divisible by 4.

The number 367744 is not divisible by 12, since it is not divisible by 3. The sum of its digits is 29, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 15

A number is divisible by 15 if it is divisible by 3 and divisible by 5. We can use each of the divisibility tests to check if a number is divisible by 15: its units digit is 0 or 5, and the sum of its digits is divisible by 3.

Example:

The number 7145580 is divisible by 15, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.

Divisibility by 16

A whole number is divisible by 16 if the number formed by the last four digits is divisible by 16.

Examples:

The number 898630032 is divisible by 16 since the number formed by its last four digits, 32, is divisible by 16.

The number 1772300000 is divisible by 16 since the number formed by its last four digits, 0, is divisible by 16.

The number 339122481296 is divisible by 16 since the number formed by its last four digits, 1296, is divisible by 16.

If a number is not divisible by 16, the remainder when the number is divided by 16 is the same as the remainder when the last four digits are divided by 16.

Example:

The number 172411045 is not divisible by 16, since the number formed by its last four digits, 1045, is not divisible by 16. When 172411045 is divided by 16, the remainder is 5, since when 1045 is divided by 16, the remainder is 5.

Divisibility by 18

A number is divisible by 18 if it is divisible by 2 and divisible by 9. We can use each of the divisibility tests to check if a number is divisible by 18: its units digit is even and the sum of its digits is divisible by 9.

Examples:

The number 7145586 is divisible by 18, since its units digit is even, and the sum of its digits is 36, which is divisible by 9.

The number 2976633 is not divisible by 18, since its units digit is not even.

The number 367942 is not divisible by 18, since it is not divisible by 9. The sum of its digits is 31, which is not divisible by 9, so the number 367942 is not divisible by 9.

Divisibility by 20

A number is divisible by 20 if its units digit is 0, and its tens digit is even. In other words, the last two digits form one of the numbers 0, 20, 40, 60, or 80.

Examples:

The number 3351002760 is divisible by 20, since the number formed by its last two digits is 60.

The number 802199730000 is divisible by 20, since the number formed by its last two digits is 0.

Divisibility by 22

A number is divisible by 22 if it is divisible by the numbers 2 and 11. We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11.

Example:

Is the number 117524 divisible by 22? The units digit is even, so it is divisible by 2. The two sums of every other digit are 4 + 5 + 1 = 10 and 2 + 7 + 1 = 10, which have a difference of 0. Since 0 is divisible by 11, 117524 is divisible by 11. Thus, 117524 is divisible by 22, since it is divisible by both 2 and 11.

Divisibility by 25

A number is divisible by 25 if the number formed by the last two digits is any of 0, 25, 50, or 75 (the number formed by its last two digits is divisible by 25).

Examples:

The number 73224050 is divisible by 25, since its last two digits form the number 50.

The number 1008922200 is divisible by 25, since its last two digits form the number 0.