the number of 3-digit numbers which end in 7 and are divisible by 11 is?

To find the number of 3-digit numbers that end in 7 and are divisible by 11, we can take the following steps:

Step 1: Determine the range of 3-digit numbers.
Since they are 3-digit numbers, the range is from 100 to 999.

Step 2: Determine the range of numbers that end in 7.
The numbers that end in 7 have a pattern where every 10th number ends in 7. So, the range of numbers ending in 7 is from 107 to 997.

Step 3: Determine the range of numbers that are divisible by 11.
The numbers that are divisible by 11 also have a pattern. They differ by a multiple of 11 between each other. So, we can start from the smallest number in the range of numbers ending in 7 (107) and keep adding 11 until we reach the highest number (997).

Step 4: Count the numbers in the range of numbers divisible by 11.
To count the numbers in the range of numbers divisible by 11, we can subtract the starting number from the ending number and divide it by 11. Then, we add 1 to include the starting number as well.

107 to 997 = 107, 118, 129, 140, ..., 986, 997

997 - 107 = 890
890 / 11 = 80

So, the range of numbers divisible by 11 is 80 numbers.

Step 5: Determine the number of 3-digit numbers divisible by 11 and ending in 7.
Within the range of numbers divisible by 11, we need to find the numbers that are 3-digit numbers. Since the starting number is 107, we need to subtract any numbers less than 100 from our count.

100 to 107 has 7 numbers less than 100.

80 - 7 = 73

So, there are 73 numbers that meet both criteria: they are 3-digit numbers, they end in 7, and they are divisible by 11.

To find the number of 3-digit numbers that end in 7 and are divisible by 11, we can follow these steps:

Step 1: Identify the range
Since we are looking for 3-digit numbers, we need to determine the range of numbers that fall under this category. In this case, the range would be from 100 to 999 (inclusive).

Step 2: Determine the first number that ends in 7 and is divisible by 11
To find the first number that meets these criteria, we can start from 100 and check each subsequent number until we find one that ends in 7 and is divisible by 11. We can do this by incrementing the number by 11 until we find a suitable number.

100 + 11 = 111
111 + 11 = 122
.
.
.
165 + 11 = 176

So, the first 3-digit number that ends in 7 and is divisible by 11 is 176.

Step 3: Determine the last number that ends in 7 and is divisible by 11
To find the last number, we need to determine the largest 3-digit number that ends in 7 and is divisible by 11. We can do this by finding the largest multiple of 11 that is smaller than 999 and ends in 7.

The largest multiple of 11 smaller than 999 is 990. However, this number does not end in 7, so we need to subtract 11 from it until we get a number that satisfies the conditions.

990 - 11 = 979
979 - 11 = 968
.
.
.
935 - 11 = 924

So, the last 3-digit number that ends in 7 and is divisible by 11 is 924.

Step 4: Calculate the number of 3-digit numbers
To find the number of 3-digit numbers that meet these criteria, we can subtract the first number from the last number and add 1 (since the first and last numbers are included in the count).

Number of 3-digit numbers = (924 - 176) + 1 = 749

Therefore, there are 749 3-digit numbers that end in 7 and are divisible by 11.

Multiples of 11 which end in 7 are

11*17 = 187
11*27 = 297
...
11*87 = 957

So, how many are there?