how many positive integers less than 1000 are not divisible by a 1 digit prime
Matematika
RadityaRizky3
Pertanyaan
how many positive integers less than 1000 are not divisible by a 1 digit prime
1 Jawaban
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1. Jawaban Je7
I will use 1 digit prime as = 2,3 and 5.
Instead of figuring out how many are not divisible by 2, 3 and 5, let’s figure out how many ARE divisible by those values.
How many numbers are divisible by 2?
1000 ÷ 2 = 500 → ADD 500
→ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20 … 996, 998, 1000}How many are divisible by 3?
1000 ÷ 3 = 333 → ADD 333
→ {3, 6, 9, 12, 15, 18, 21, 24, 27, 30 … 993, 996, 999}How many are divisible by 5?
1000 ÷ 5 = 200 → ADD 200
→ {5, 10, 15, 20, 25, 30, 35, 40 … 990, 995, 1000}
BUT 500+333+200 = 1033, more numbers than we started with.We have counted the multiples of 6 twice, so we have to subtract the overlap.
How many numbers are multiples of 6?
1000 ÷ 6 = 166 → SUBTRACT 166
→ {6, 12, 18, 24, 30, 36, 42, 48 … 984, 990, 996}We have counted the multiples of 10 twice, so we have to subtract the overlap
How many numbers are multiples of 10?
1000 ÷ 10 = 100 → SUBTRACT 100
→ {10, 20, 30, 40, 50, 60, 70 … 980, 990, 1000}We have counted the multiples of 15 twice, so we have to subtract the overlap
How many numbers are multiples of 15?
1000 ÷ 15 = 66 → SUBTRACT 66
→ {15, 30, 45, 60, 75, 90, 105, 120 … 960, 975, 990}
BUT we are not done yet: 1033 - 166 - 100 - 66 = 701
We subtracted the multiples of 30 three times, when we should have only subtracted them twice, so we need to add the triple overlap back to our sum.
How many numbers are multiple of 30?
1000 ÷ 30 = 33 → ADD + 33
→ {30, 60, 90, 120, 150, 180, 210 … 930, 960, 990}We now have 701 + 33 = 734 numbers that are multiples of 2, 3 or 5.
How many numbers between 1 and 1000 are not multiples of these three numbers? Subtract that sum from 1000 and what do you get?
Here they are:
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