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We need to find the numbers between 1 and 21,000,000 that are divisible by 2, 3, or 5, but not by 7. This means we will first count those divisible by 2, 3, or 5 including 1 to 21,000,000.

Using the inclusion-exclusion principle, count the numbers divisible by 2, 3, or 5:- Numbers divisible by 2: \[ \left\lfloor \frac{21,000,000}{2} \right\rfloor = 10,500,000 \]- Numbers divisible by 3: \[ \left\lfloor \frac{21,000,000}{3} \right\rfloor = 7,000,000 \]- Numbers divisible by 5: \[ \left\lfloor \frac{21,000,000}{5} \right\rfloor = 4,200,000 \]

To adjust for overcounting, subtract numbers counted twice due to being divisible by combinations of these numbers:- Divisible by 6 (2 and 3): \[ \left\lfloor \frac{21,000,000}{6} \right\rfloor = 3,500,000 \]- Divisible by 10 (2 and 5): \[ \left\lfloor \frac{21,000,000}{10} \right\rfloor = 2,100,000 \]- Divisible by 15 (3 and 5): \[ \left\lfloor \frac{21,000,000}{15} \right\rfloor = 1,400,000 \]

Add back numbers counted three times (as they have been subtracted more than necessary) and which are divisible by 30 (2, 3, and 5):- Divisible by 30: \[ \left\lfloor \frac{21,000,000}{30} \right\rfloor = 700,000 \]

Combine previous inclusions and exclusions:\[ 10,500,000 + 7,000,000 + 4,200,000 - 3,500,000 - 2,100,000 - 1,400,000 + 700,000 = 15,400,000 \]

Now, subtract from this the numbers divisible by 2 or 3 or 5 that are also divisible by 7. We apply the same inclusion-exclusion principle here:- Divisible by 14 (2 and 7): \[ \left\lfloor \frac{21,000,000}{14} \right\rfloor = 1,500,000 \]- Divisible by 21 (3 and 7): \[ \left\lfloor \frac{21,000,000}{21} \right\rfloor = 1,000,000 \]- Divisible by 35 (5 and 7): \[ \left\lfloor \frac{21,000,000}{35} \right\rfloor = 600,000 \]- Add those divisible by 42, 70, and 105 (2, 3, 5, and 7): * Divisible by 42: \[ \left\lfloor \frac{21,000,000}{42} \right\rfloor = 500,000 \] * Divisible by 70: \[ \left\lfloor \frac{21,000,000}{70} \right\rfloor = 300,000 \] * Divisible by 105: \[ \left\lfloor \frac{21,000,000}{105} \right\rfloor = 200,000 \]- Divisible by 210 (2, 3, 5, and 7): \[ \left\lfloor \frac{21,000,000}{210} \right\rfloor = 100,000 \]

The count of numbers divisible by 2, 3, or 5 but not 7 is:\[ 15,400,000 - (1,500,000 + 1,000,000 + 600,000 - 500,000 - 300,000 - 200,000 + 100,000) \]Simplifying the expression inside the parentheses:\[ 1,500,000 + 1,000,000 + 600,000 - 500,000 - 300,000 - 200,000 + 100,000 = 2,200,000 \]Thus, the final result is:\[ 15,400,000 - 2,200,000 = 13,200,000 \]