An odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.
In Spanish
número impar
Also called
odd integer, odd
Grade level
K-5
Grade 2Grade 3Grade 4CCSS 2.OA.C.3TABEGEDSATACT
Odd Number in a picture
Pair things up: if one is left over, the number is odd. That leftover one is the + 1 in 2k + 1.
Formal definition and notation
An integer n is odd if there is an integer k with n = 2k + 1; equivalently, n is not divisible by 2, so n ÷ 2 has remainder 1. The odd integers are …, −5, −3, −1, 1, 3, 5, …. In base ten, an integer is odd exactly when its ones digit is 1, 3, 5, 7 or 9. Integers that are not odd are even, and 0 is even.
Symbol
Read it as
Meaning
2k + 1
two k plus one
the general form of an odd number, for any integer k
2k − 1
two k minus one
another way to write odd numbers; k = 1 gives 1
Key facts
Odd and even numbers alternate: 1 odd, 2 even, 3 odd, 4 even, and so on.
odd + odd = even (3 + 5 = 8); odd + even = odd (3 + 4 = 7).
odd × odd = odd (3 × 5 = 15); anything × even = even (3 × 4 = 12).
Many primes are odd, but odd does not mean prime: 9, 15 and 21 are odd and composite. And 2 is prime but even.
Quick check: 3 questions
Question 1 of 3
Which number is odd?
Show answer and explanation
Answer: A. 3,457 ends in 7, so it is odd. The others end in 8 or 0 and are even. Only the last digit matters.
Question 2 of 3
What happens when you multiply two odd numbers?
Show answer and explanation
Answer: B.(2a + 1)(2b + 1) = 2(2ab + a + b) + 1, which is odd. Examples: 3 × 5 = 15, 7 × 9 = 63. A product is even only when at least one factor is even.
Question 3 of 3
If n is any integer, which expression is always odd?
Show answer and explanation
Answer: C.2n is always even, so 2n + 1 is always odd. n + 1 and 3n are odd only for some n (try n = 2 and n = 1).
Odd Number: frequently asked questions
Is 0 an odd number?
No, 0 is even. A number is even when dividing by 2 leaves no remainder, and 0 ÷ 2 = 0 exactly. 0 also fits the pattern on the number line: −1 is odd, 0 is even, 1 is odd. Odd numbers are the integers of the form 2k + 1, and 0 is not one of them.
Is 1 an odd number?
Yes. 1 is the smallest positive odd number: 1 = 2 × 0 + 1, and dividing 1 by 2 leaves remainder 1. The odd numbers start 1, 3, 5, 7, 9 and continue by adding 2 each time. Do not confuse this with primes: 1 is odd but it is not prime.
How can you tell if a big number is odd?
Look only at the last digit. If it is 1, 3, 5, 7 or 9, the number is odd; if it is 0, 2, 4, 6 or 8, it is even. This works because every other place value is a multiple of 10, which is even, so only the ones digit can leave a remainder when you divide by 2.
MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8Consecutive IntegersConsecutive integers are integers that follow each other in order, each 1 more than the one before, such as 7, 8, 9 or −2, −1, 0. In algebra, write them as n, n + 1, n + 2. Consecutive even or consecutive odd integers are 2 apart: n, n + 2, n + 4.Algebra · Grade 7, Grade 8, Algebra 1FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6Composite NumberA composite number is a whole number greater than 1 that has more than two factors. It can be divided evenly by a number other than 1 and itself: 12 is composite because 1, 2, 3, 4, 6 and 12 all divide it. Every whole number above 1 is either prime or composite; 1 is neither.Number System · Grade 4, Grade 5, Grade 6