There are 26 two-digit numbers that are divisible by 3 but not by 7.
Here's how we arrive at that answer:
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Identify the range of two-digit numbers: The two-digit numbers range from 10 to 99.
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Find the number of two-digit numbers divisible by 3:
- The first two-digit number divisible by 3 is 12.
- The last two-digit number divisible by 3 is 99.
- To find the total count, we can use the formula for arithmetic sequences: (Last term - First term)/Common difference + 1. So, (99 - 12)/3 + 1 = 87/3 + 1 = 29 + 1 = 30. Therefore, there are 30 two-digit numbers divisible by 3.
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Find the number of two-digit numbers divisible by both 3 and 7: A number divisible by both 3 and 7 is divisible by their least common multiple, which is 21.
- The first two-digit number divisible by 21 is 21.
- The last two-digit number divisible by 21 is 84.
- Numbers divisible by 21: 21, 42, 63, 84.
- Therefore, there are 4 two-digit numbers divisible by both 3 and 7 (i.e., divisible by 21).
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Subtract the numbers divisible by both 3 and 7 from the numbers divisible by 3: To find the number of two-digit numbers divisible by 3 but not by 7, we subtract the count of numbers divisible by both from the count of numbers divisible by 3.
- 30 (divisible by 3) - 4 (divisible by both 3 and 7) = 26.
Therefore, there are 26 two-digit numbers divisible by 3 but not by 7.
