Counting 2-Digit Positive Integers Divisible by 4 or 9
You’ve come to the correct site if you’ve ever wondered how many positive integers with two digits that can be divided by four or nine. We will examine this query in this blog post and show you a methodical way to discover the response.
Divisibility Rules
Allow me to briefly go over the divisibility rules for 4 and 9 before we get started with the computations. We can use these criteria to find out if a given integer divides into either of these two numbers.
If the final two digits of a number result in a multiple of 4, then the value is divisible by 4. For instance, since 48 is a multiple of 4, it is divisible by 4. Nevertheless, since 57 is not a multiple of 4, it cannot be divided by 4.
If a number’s total of its digits is a multiple of nine, then it is divisible by nine. For instance, since 6 + 3 = 9 is a multiple of 9, the number 63 is divisible by 9. But since 4 + 7 = 11 is not a multiple of 9, the number 47 is not divisible by 9.
Counting 2-Digit Integers Divisible by 4
Counting the number of 2-digit positive integers that are divisible by 4 is a good place to start. Finding the range of 2-digit integers and counting how many of them are divisible by 4 are the first steps in doing this.
10 is the smallest two-digit number while 99 is the greatest. We must count the multiples of 4 that fall inside this range in order to determine the total number of integers that are divisible by 4.
To begin, let’s see how small a multiple of 4 may be that still equals or exceeds 10. We can determine that 12 is the first two-digit positive integer divisible by 4 because it is the lowest multiple of 4.
Finding the largest multiple of 4 that is less than or equal to 99 is the next task. 96 is the biggest multiple of 4 that is less than or equal to 99. Consequently, we can conclude that the final two-digit positive integer divisible by four is 96.
By deducting the first integer from the last and adding one, we can get the total count now that we know the first and last two-digit integers that are divisible by four. The count in this instance is 96 – 12 + 1 = 85.
Counting 2-Digit Integers Divisible by 9
Let’s now count how many positive integers with two digits that can be divided by nine. We’ll take a similar route as when we counted the integers that were divisible by 4.
Eighteen is the smallest multiple of nine that is either bigger than or equal to ten. Consequently, the first two-digit positive integer divisible by nine is eighteen.
99 is the greatest multiple of 9 that is either equal to or less than 99. Therefore, the final two-digit positive integer divisible by nine is 99.
We can obtain the count by subtracting the first integer from the last integer and adding one using the same technique as previously. The count in this instance is 99 – 18 + 1 = 82.
Counting 2-Digit Integers Divisible by 4 or 9
Let’s count the number of positive integers with two digits that can be divided by four or nine. In order to accomplish this, we must first determine which numbers are divisible by 4 and which by 9, then subtract the overlap.
We know that there are 82 numbers divisible by 9 and 85 integers divisible by 4, based on our earlier computations. But we have to deduct the overlap, which is the part of the numbers that divides by 4 and 9.
We must determine the multiples of the 36 least common multiple (LCM) between 4 and 9 in order to determine the overlap. 36 alone is the lowest multiple of 36 that is either bigger than or equal to 10. 72 is the greatest multiple of 36 that is either equal to or less than 99.
The number of 2-digit positive integers that are divisible by 4 or 9 is therefore (85 + 82) – (72 – 36 + 1) = 130.