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18 Mar 2019, 02:40
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
33% (02:55) correct
67%
(02:55)
wrong
based on 97
sessions
History
Date
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How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?
(A) 768
(B) 801
(C) 934
(D) 1067
(E) 1167
(A) 768
(B) 801
(C) 934
(D) 1067
(E) 1167
Archit3110
18 Mar 2019, 02:51
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Bunuel
2001/3 = 667
2001/4 = 500
2001/12 = 166
2001/15 = 133
2001/20 =100
2001/60 = 33
667+500-166= 1001
133+100-33 = 200
1001-200; 801
IMO B
General Discussion
25 Sep 2020, 03:54
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By the counting principle, the number of multiples between 2 numbers is given by the below equation
\(\frac{Largest \space Value - Smallest \space Value}{Divisor}\) + 1, where The divisor is the number whose multiples we need.
Multiples of 3 or 4 but not 5. M(3) + M(4) - Multiples of LCM(3,4) - Multiples of LCM(3,5) - Multiples of LCM(4,5) + Multiples of LCM(3,4,5)
For Multiples of 3.
The smallest number divisible by 3 = 3
The largest number divisible by 3 = 2001
Number of multiples of 3 = \(\frac{2001 - 3}{3} + 1 = \frac{1998}{3} + 1 = 666 + 1 = 667\)
For Multiples of 4.
The smallest number divisible by 4 = 4
The largest number divisible by 4 = 2000
Number of multiples of 4 = \(\frac{2000 - 4}{4} + 1 = \frac{1996}{4} + 1 = 499 + 1 = 500\)
For Multiples of LCM(3,4) = 12.
The smallest number divisible by 12 = 12
The largest number divisible by 12 = 1992
Number of multiples of 12 = \(\frac{1992 - 12}{12} + 1 = \frac{1980}{12} + 1 = 165 + 1 = 166\)
For Multiples of LCM(3,5) = 15.
The smallest number divisible by 15 = 15
The largest number divisible by 15 = 1995
Number of multiples of 15 = \(\frac{1995 - 15}{15} + 1 = \frac{1980}{15} + 1 = 132 + 1 = 133\)
For Multiples of LCM(4,5) = 20.
The smallest number divisible by 20 = 20
The largest number divisible by 20 = 2000
Number of multiples of 20 = \(\frac{2000 - 20}{20} + 1 = \frac{1980}{20} + 1 = 99 + 1 = 100\)
For Multiples of LCM(3,4,5) = 60.
The smallest number divisible by 60 = 60
The largest number divisible by 60 = 1980
Number of multiples of 60 = \(\frac{1980 - 60}{60} + 1 = \frac{1920}{60} + 1 = 32 + 1 = 33\)
Therefore, the number of Multiples = 667 + 500 - 166 - 133 - 100 + 33 = 801
Option B
Arun Kumar
\(\frac{Largest \space Value - Smallest \space Value}{Divisor}\) + 1, where The divisor is the number whose multiples we need.
Multiples of 3 or 4 but not 5. M(3) + M(4) - Multiples of LCM(3,4) - Multiples of LCM(3,5) - Multiples of LCM(4,5) + Multiples of LCM(3,4,5)
For Multiples of 3.
The smallest number divisible by 3 = 3
The largest number divisible by 3 = 2001
Number of multiples of 3 = \(\frac{2001 - 3}{3} + 1 = \frac{1998}{3} + 1 = 666 + 1 = 667\)
For Multiples of 4.
The smallest number divisible by 4 = 4
The largest number divisible by 4 = 2000
Number of multiples of 4 = \(\frac{2000 - 4}{4} + 1 = \frac{1996}{4} + 1 = 499 + 1 = 500\)
For Multiples of LCM(3,4) = 12.
The smallest number divisible by 12 = 12
The largest number divisible by 12 = 1992
Number of multiples of 12 = \(\frac{1992 - 12}{12} + 1 = \frac{1980}{12} + 1 = 165 + 1 = 166\)
For Multiples of LCM(3,5) = 15.
The smallest number divisible by 15 = 15
The largest number divisible by 15 = 1995
Number of multiples of 15 = \(\frac{1995 - 15}{15} + 1 = \frac{1980}{15} + 1 = 132 + 1 = 133\)
For Multiples of LCM(4,5) = 20.
The smallest number divisible by 20 = 20
The largest number divisible by 20 = 2000
Number of multiples of 20 = \(\frac{2000 - 20}{20} + 1 = \frac{1980}{20} + 1 = 99 + 1 = 100\)
For Multiples of LCM(3,4,5) = 60.
The smallest number divisible by 60 = 60
The largest number divisible by 60 = 1980
Number of multiples of 60 = \(\frac{1980 - 60}{60} + 1 = \frac{1920}{60} + 1 = 32 + 1 = 33\)
Therefore, the number of Multiples = 667 + 500 - 166 - 133 - 100 + 33 = 801
Option B
Arun Kumar
