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Find out how many 2-digit numbers can be divided by 4 or 9. Our exploration will simplify the math and show you the interesting patterns behind these divisible numbers.
How many 2 – digit Positive Integers are divisible by 4 or 9?
To find the number of 2-digit positive integers divisible by 4 or 9, we’ll use the principle of inclusion-exclusion.
Counting multiples of 4:
- The smallest 2-digit multiple of 4 is 12.
- The largest 2-digit multiple of 4 is 96.
- This forms an arithmetic progression with a common difference of 4.
- The number of terms in this progression is (96 – 12) / 4 + 1 = 22.
Counting multiples of 9:
- The smallest 2-digit multiple of 9 is 18.
- The largest 2-digit multiple of 9 is 99.
- This forms an arithmetic progression with a common difference of 9.
- The number of terms in this progression is (99 – 18) / 9 + 1 = 10.
Counting multiples of both 4 and 9 (to avoid overcounting):
- These are the multiples of 36 (the least common multiple of 4 and 9).
- The smallest 2-digit multiple of 36 is 36 itself.
- The largest 2-digit multiple of 36 is 72.
- There are only 2 such numbers.
Combining the counts:
- Total count = (count of multiples of 4) + (count of multiples of 9) – (count of multiples of both 4 and 9)
- Total count = 22 + 10 – 2 = 30
Therefore, there are 30 two-digit positive integers that are divisible by 4 or 9.
What are Integers?
Integers are a set of whole numbers that do not have any fractional or decimal parts. They include positive numbers, negative numbers, and zero. In mathematical notation, integers are represented by the symbol “Z.” The set of integers includes all the counting numbers (1, 2, 3, …) along with their negatives (-1, -2, -3, …) and zero (0).
Mathematically, integers can be expressed as {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers are used in various mathematical operations, and they form the basis for many mathematical concepts and calculations.
Examples of integers:
- Positive integers: 1, 2, 3, 4, …
- Negative integers: -1, -2, -3, -4, …
- Zero: 0
Integers are fundamental in algebra, number theory, and many branches of mathematics. They are also commonly used in computer programming and various scientific and engineering applications.
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