Question

The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ , is_________

Answer 1

Correct Answer: 514

To find the number of 3-digit numbers divisible by either 2 or 3, but not by 7, we follow these steps:

Step 1: Calculate the total number of 3-digit numbers.

Three-digit numbers range from 100 to 999, inclusive:

Total 3-digit numbers = 999 - 100 + 1 = 900

Step 2: Find the count of 3-digit numbers divisible by 2.

The sequence of numbers divisible by 2 starts with 100 and ends with 998:

Count = (998 - 100)/2 + 1 = 450

Step 3: Find the count of 3-digit numbers divisible by 3.

The sequence starts with 102 and ends with 999:

Count = (999 - 102)/3 + 1 = 300

Step 4: Use the principle of inclusion-exclusion to find numbers divisible by either 2 or 3.

Calculate the count of 3-digit numbers divisible by both 2 and 3 (i.e., divisible by 6).

Numbers divisible by 6 start with 102 and end with 996:

Count = (996 - 102)/6 + 1 = 150

Using inclusion-exclusion, numbers divisible by 2 or 3:

Count = 450 + 300 - 150 = 600

Step 5: Subtract numbers divisible by 7 from those calculated in Step 4.

a. Numbers divisible by both 2 and 7 (14):

Count = (994 - 112)/14 + 1 = 64

b. Numbers divisible by both 3 and 7 (21):

Count = (987 - 105)/21 + 1 = 43

c. Numbers divisible by 2, 3, and 7 (42):

Count = (966 - 126)/42 + 1 = 21

Using inclusion-exclusion for numbers divisible by 7:

Count = 64 + 43 - 21 = 86

Finally, subtract 86 from 600 to get the final count:

Final Count = 600 - 86 = 514

The number of 3-digit numbers divisible by either 2 or 3 but not 7 is 514. This result fits within the given range of 514,514.