please teach 12 years old student how to solve this question in visual animation and related knowledge in General Strategy for Two-Rule Filtering Problems in each steps: Donna likes all numbers that are not divisible by 3 and Sandra likes all numbers
that have no digits divisible by 3. How many four-digit numbers are there that
both Donna and Sandra like?
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Let's understand this problem step by step. Donna likes all numbers that are NOT divisible by 3, while Sandra likes numbers that have NO digits divisible by 3. We need to find four-digit numbers from 1000 to 9999 that BOTH Donna and Sandra like. This is a two-rule filtering problem where we need numbers satisfying both conditions simultaneously.
Let's understand Donna's rule in detail. A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 1247 has digit sum 1+2+4+7=14, which gives remainder 2 when divided by 3, so Donna likes it. But 1236 has digit sum 12, which is divisible by 3, so Donna doesn't like it. We can use this digit sum rule to quickly check any four-digit number.
Now let's understand Sandra's rule. Sandra checks each individual digit to see if it's divisible by 3. The forbidden digits are 3, 6, and 9, while the allowed digits are 0, 1, 2, 4, 5, 7, and 8. For example, 1247 has all allowed digits, so Sandra likes it. But 1236 contains the digits 3 and 6, which are divisible by 3, so Sandra rejects this number.
For two-rule filtering problems, we use a systematic strategy. First, apply the more restrictive rule to eliminate as many numbers as possible. Sandra's rule is more restrictive because it eliminates any number containing the digits 3, 6, or 9. We start with 9000 four-digit numbers, apply Sandra's filter to get 2058 numbers, then apply Donna's filter to get our final answer of 1372 numbers.
Now let's count how many four-digit numbers Sandra likes. For the first digit, we can't use 0 because it wouldn't be a four-digit number, and we can't use 3, 6, or 9 because of Sandra's rule. So we have 6 choices: 1, 2, 4, 5, 7, 8. For each of the remaining three digits, we can use any digit except 3, 6, 9, giving us 7 choices each including 0. The total is 6 times 7 times 7 times 7, which equals 2058 numbers that Sandra likes.