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16 Sep 2014, 01:15
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How many integers between 18 and 70, inclusive, are divisible by 7 or 9, but not by both ?
A. 11
B. 12
C. 13
D. 14
E. 15
A. 11
B. 12
C. 13
D. 14
E. 15
16 Sep 2014, 01:15
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Bookmarks
Official Solution:
How many integers between 18 and 70, inclusive, are divisible by 7 or 9, but not by both ?
A. 11
B. 12
C. 13
D. 14
E. 15
The number of multiples of an integer in a range is given by the formula \(\frac{\text{last multiple in the range - first multiple in the range}}{\text{multiple}}+1\). Hence:
• Number of multiples of 7 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{(70-21)}{7}+1=8\);
• Number of multiples of 9 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{(63-18)}{9}+1=6\);
• Number of multiples of both 7 and 9 is 1, which is \(7*9=63\).
Note that the number 63 is a multiple of both 7 and 9 and is included in the count of multiples of 7 and the count of multiples of 9. Since we don't want it to be counted at all, we need to subtract it once from the total count of multiples of 7, and once from the total count of multiples of 9.
Therefore, the number of positive integers that are either multiples of 7 or multiples of 9, but not multiples of both, in the given range is \((8-1)+(6-1)=12\).
Answer: B
How many integers between 18 and 70, inclusive, are divisible by 7 or 9, but not by both ?
A. 11
B. 12
C. 13
D. 14
E. 15
The number of multiples of an integer in a range is given by the formula \(\frac{\text{last multiple in the range - first multiple in the range}}{\text{multiple}}+1\). Hence:
• Number of multiples of 7 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{(70-21)}{7}+1=8\);
• Number of multiples of 9 in the given range \(\frac{(\text{last-first})}{\text{multiple}}+1=\frac{(63-18)}{9}+1=6\);
• Number of multiples of both 7 and 9 is 1, which is \(7*9=63\).
Note that the number 63 is a multiple of both 7 and 9 and is included in the count of multiples of 7 and the count of multiples of 9. Since we don't want it to be counted at all, we need to subtract it once from the total count of multiples of 7, and once from the total count of multiples of 9.
Therefore, the number of positive integers that are either multiples of 7 or multiples of 9, but not multiples of both, in the given range is \((8-1)+(6-1)=12\).
Answer: B
