How many integers exist which leave a remainder of 2 when divided by 3?
Answer (2)
There are infinitely many integers that leave a remainder of 2 when divided by 3. These integers can be expressed in the form of n = 3 k + 2 , where k is any integer. Therefore, you can have positive, negative, or zero integers such as 2, 5, -1, etc.
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The question asks how many integers exist which leave a remainder of 2 when divided by 3. To solve this, let's first understand what it means for a number to leave a remainder of 2 when divided by 3.
When you divide an integer by 3, there are three possible remainders: 0, 1, or 2. If a number leaves a remainder of 2, it means that when this number is divided by 3, the result is not a whole number, but rather something of the form n + 3 2 , where n is an integer.
For example:
The number 5 divided by 3 gives a quotient of 1 and a remainder of 2, since 5 = 3 × 1 + 2 .
The number 8 divided by 3 gives a quotient of 2 and a remainder of 2, since 8 = 3 × 2 + 2 .
The numbers that fit this description can be expressed in the general form:
x = 3 k + 2
where k is any integer. This expression shows that as k varies over all integers, x takes on all possible integers that leave a remainder of 2 when divided by 3.
Now, considering the set of all integers, there is no upper or lower limit on k . This means k as an integer can be any number from negative infinity to positive infinity.
Therefore, the number of integers leaving a remainder of 2 when divided by 3 is infinite. In mathematics, infinite sets are aligned with the idea that they cannot be measured in size the same way finite sets can be, hence we say there are infinitely many such integers.
