Understanding the Problem: Two-Digit Numbers Divisible by 4

The question asks us to find the total count of two-digit whole numbers that can be perfectly divided by 4. A two-digit number is any integer from 10 up to 99, inclusive.

To be divisible by 4 means that when you divide the number by 4, the remainder is zero. We are looking for numbers 'N' such that 10 <= N <= 99 and N is a multiple of 4.

Identifying the Range of Two-Digit Multiples of 4

First, let's find the smallest two-digit number that is divisible by 4. We start checking from 10:

• 10 ÷ 4 = 2 with a remainder.
• 11 ÷ 4 = 2 with a remainder.
• 12 ÷ 4 = 3 with no remainder.

So, the smallest two-digit number divisible by 4 is 12.

Next, let's find the largest two-digit number that is divisible by 4. We start checking from 99, going downwards:

• 99 ÷ 4 = 24 with a remainder of 3.
• 98 ÷ 4 = 24 with a remainder of 2.
• 97 ÷ 4 = 24 with a remainder of 1.
• 96 ÷ 4 = 24 with no remainder.

So, the largest two-digit number divisible by 4 is 96.

The two-digit numbers divisible by 4 form a sequence: 12, 16, 20, ..., 96. This is an arithmetic progression where each term is obtained by adding 4 to the previous term.

Calculating the Count of Two-Digit Numbers Divisible by 4

We have an arithmetic sequence:

• First term (\(a_1\)): 12
• Common difference (\(d\)): 4
• Last term (\(a_n\)): 96

We want to find the number of terms (\(n\)) in this sequence. The formula for the n-th term of an arithmetic progression is:

a_n = a_1 + (n-1)d

Substitute the values we know into the formula:

96 = 12 + (n-1)4

Now, solve for \(n\):

96 - 12 = (n-1)4

84 = (n-1)4

Divide both sides by 4:

\frac{84}{4} = n-1

21 = n-1

Add 1 to both sides:

n = 21 + 1

n = 22

There are 22 two-digit numbers that are divisible by 4.

Alternative Method: Using Multiples

Another way to think about this is to find which multiples of 4 fall within the two-digit range (10-99).

We are looking for numbers of the form 4 \times k, where k is an integer, such that:

10 \le 4k \le 99

To find the range for k, divide the inequality by 4:

\frac{10}{4} \le k \le \frac{99}{4}

2.5 \le k \le 24.75

Since k must be an integer (representing the factor of 4), the possible integer values for k start from the smallest integer greater than or equal to 2.5, which is 3, and go up to the largest integer less than or equal to 24.75, which is 24.

So, the integer values of k are 3, 4, 5, ..., 24.

To find the count of these integers, subtract the smallest value from the largest value and add 1 (because we include both ends):

Count = (Largest value of k) - (Smallest value of k) + 1

Count = 24 - 3 + 1

Count = 21 + 1

Count = 22

Both methods confirm that there are 22 two-digit numbers divisible by 4.

Revision Table: Counting Multiples

Additional Information: Divisibility Rule for 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For two-digit numbers, the number itself is formed by its last two digits.

Examples:

• 12: Last two digits form 12. 12 is divisible by 4. So, 12 is divisible by 4.
• 36: Last two digits form 36. 36 is divisible by 4 (36 = 4 × 9). So, 36 is divisible by 4.
• 96: Last two digits form 96. 96 is divisible by 4 (96 = 4 × 24). So, 96 is divisible by 4.
• 43: Last two digits form 43. 43 is not divisible by 4 (43 = 4 × 10 + 3). So, 43 is not divisible by 4.

This rule helps in quickly checking individual numbers but finding the count requires a systematic approach like the ones described above, identifying the first and last multiples in the range.