Of-the-numbers-1-2-3-6000-how-many-are-not-multiples-of-2-3-or-5-

Question Number 2381 by Yozzi last updated on 18/Nov/15

O f t h e n u m b e r s 1, 2, 3, \dots, 6000,

h o w m a n y a r e n o t m u l t i p l e s o f 2, 3 o r 5 ?

Commented by 123456 last updated on 18/Nov/15

N = N_{2} + N_{3} + N_{5} - N_{2, 3} - N_{2, 5} - N_{3, 5} + N_{2, 3, 5}

M = X - N

Commented by prakash jain last updated on 18/Nov/15

n (A \cup B \cup C) = n (A) + n (B) + n (C)

- n (A \cap B) - n (A \cap C) - n (B \cap C)

+ n (A \cap B \cap C)

Answered by 123456 last updated on 18/Nov/15

first we have that 1, 2, \dots, 6000 have

6000 numer (X = 6000)

lets count the number of multiples of

2, 3, 5, for this, we only need to count

the number of multiply of these and

subtract tese that are multply of two

and sum the multiple of tree of these

ps : - ⌈ \frac{1}{a} ⌉ + 1 = 0, a \in N^{*}

N_{2} = ⌊ \frac{6000}{2} ⌋ = 3000

N_{3} = ⌊ \frac{6000}{3} ⌋ = 2000

N_{5} = ⌊ \frac{6000}{5} ⌋ = 1200

N_{2, 3} = ⌊ \frac{6000}{2 \times 3} ⌋ = ⌊ \frac{6000}{6} ⌋ = 1000

N_{2, 5} = ⌊ \frac{6000}{2 \times 5} ⌋ = ⌊ \frac{6000}{10} ⌋ = 600

N_{3, 5} = ⌊ \frac{6000}{3 \times 5} ⌋ = ⌊ \frac{6000}{15} ⌋ = 400

N_{2, 3, 5} = ⌊ \frac{6000}{2 \times 3 \times 5} ⌋ = ⌊ \frac{6000}{30} ⌋ = 200

N = 3000 + 2000 + 1200 - 1000 - 600 - 400 + 200

N = 4400

so, the number of no multiple is the

total minus the number of multiples

M = 6000 - 4400 = 1600

Commented by Yozzi last updated on 18/Nov/15

T h a n k s . I g o t t h a t .

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