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A certain clock marks every hour by striking a number of tim [#permalink]

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23 Mar 2010, 14:29

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110. A certain clock marks every hour by striking a number of times qual to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is 22 secs. At 12:00, how many seconds elapse between the beginning of the first stroke and the end of the last stroke?

A. 72 B. 50 C. 48 D. 46 E. 44

157. A certain right triangle has sides of length x, y, z, where x < y< z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

A. \(y> \sqrt{2}\) B. \(\frac{\sqrt{3}}{2} < y< \sqrt{2}\) C. \(\frac{\sqrt{2}}{3} < y < \frac{\sqrt{3}}{2}\) D. \(\frac{\sqrt{2}}{4} <y < \frac{\sqrt{2}}{3}\) E. \(y < \frac{\sqrt{3}}{4}\)

160. If n is a positive integer and N^2 is divisible by 72, then the largest positive integer that must divide N is

110. A certain clock marks every hour by striking a number of times qual to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is 22 secs. At 12:00, how many seconds elapse between the beginning of the first stroke and the end of the last stroke?

A. 72 B. 50 C. 48 D. 46 E. 44

at 6, the number of strokes will be 6 and the number of time interval between strokes will be 5, for a total of 11 - so each one is 2 sec. at 12, number of strokes will be 12 and the number of time intervals will be 11 for a total of 23 total time = 23 * 2 = 46

A certain right triangle has sides of length x, y, z, where x < y< z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

The area of the triangle is \(\frac{xy}{2}=1\) (\(x<y<z\) means that hypotenuse is \(z\)) --> \(x=\frac{2}{y}\). As \(x<y\), then \(\frac{2}{y}<y\) --> \(2<y^2\) --> \(\sqrt{2}<y\).

Also note that max value of \(y\) is not limited at all. For example \(y\) can be \(1,000,000\) and in this case \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).

If n is a positive integer and N^2 is divisible by 72, then the largest positive integer that must divide N is

A. 6 B. 12 C. 24 D. 36 E. 48

The largest positive integer that must divide \(n\), means for lowest value of \(n\) which satisfies the given statement in the stem. The lowest square of integer, which is multiple of \(72\) is \(144\) --> \(n^2=144\) --> \(n=12\). Largest factor of \(12\) is \(12\).

OR:

Given: \(72k=n^2\), where \(k\) is an integer \(\geq1\) (as \(n\) is positive).

\(72k=n^2\) --> \(n=6\sqrt{2k}\), as \(n\) is an integer \(\sqrt{2k}\), also must be an integer. The lowest value of \(k\), for which \(\sqrt{2k}\) is an integer is when \(k=2\) --> \(\sqrt{2k}=\sqrt{4}=2\) --> \(n=6\sqrt{2k}=6*2=12\)

If n is a positive integer and N^2 is divisible by 72, then the largest positive integer that must divide N is

A. 6 B. 12 C. 24 D. 36 E. 48

The largest positive integer that must divide \(n\), means for lowest value of \(n\) which satisfies the given statement in the stem. The lowest square of integer, which is multiple of \(72\) is \(144\) --> \(n^2=144\) --> \(n=12\). Largest factor of \(12\) is \(12\).

OR:

Given: \(72k=n^2\), where \(k\) is an integer \(\geq1\) (as \(n\) is positive).

\(72k=n^2\) --> \(n=6\sqrt{2k}\), as \(n\) is an integer \(\sqrt{2k}\), also must be an integer. The lowest value of \(k\), for which \(\sqrt{2k}\) is an integer is when \(k=2\) --> \(\sqrt{2k}=\sqrt{4}=2\) --> \(n=6\sqrt{2k}=6*2=12\)

Answer: B.

Bunuel, 48^2 is also divisible by 72. Why can't 48 be N ?
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110 i dont understand what the interval means does it mean the how many strokes to get to the 6th stroke not counting the beginning.

157. i need more questions like this, i'm not quite conveying it. i need elementary explanations with pictures

sorry! thank you!

Interval is the time delay between each stroke which is equal to the time required for stroke . The 22 seconds includes the time from the beginning of the first stroke till the end of the 6th stroke including the gap time in between.

If let's say S is the time for each stroke and I is the interval time in between S+I+S+I+S+I+S+I+S+I+S =22sec But S=I 11S =11I=22 and S=I=2 seconds

At 12:00 clock strikes 12 times and there will be 11 intervals in total between the 12 strokes so total time will be (12+11)*2 = 46

Hope this helps

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If n is a positive integer and N^2 is divisible by 72, then the largest positive integer that must divide N is

A. 6 B. 12 C. 24 D. 36 E. 48

The largest positive integer that must divide \(n\), means for lowest value of \(n\) which satisfies the given statement in the stem. The lowest square of integer, which is multiple of \(72\) is \(144\) --> \(n^2=144\) --> \(n=12\). Largest factor of \(12\) is \(12\).

OR:

Given: \(72k=n^2\), where \(k\) is an integer \(\geq1\) (as \(n\) is positive).

\(72k=n^2\) --> \(n=6\sqrt{2k}\), as \(n\) is an integer \(\sqrt{2k}\), also must be an integer. The lowest value of \(k\), for which \(\sqrt{2k}\) is an integer is when \(k=2\) --> \(\sqrt{2k}=\sqrt{4}=2\) --> \(n=6\sqrt{2k}=6*2=12\)

Answer: B.

Bunuel, 48^2 is also divisible by 72. Why can't 48 be N ?

The question is about the largest integer that must divide n. With the known information we can only say that n is divisible by 12 irrespective of the value of k. 36 and 48 also can divide n but they are dependent on k value being a multiple of 18 and 32 respectively. But since we don't know k value 12 is the largest that must divide n

Thanks
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110. A certain clock marks every hour by striking a number of times qual to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is 22 secs. At 12:00, how many seconds elapse between the beginning of the first stroke and the end of the last stroke?

solution:- at 6'o clock there will be 6 strokes, now will we find number of intervals between these 6 strokes which will be one less than total strokes i.e.5

now given in ques is

"the time required for a stroke is exactly equal to the time interval between strokes."

so total time lapsed for 1stroke (x) at 6 o clock comes out to be

(6strokes+5 interval)x = 22 x=2

now at 12o clock, there will be 12 strokes and 11 intervals i.e. total 23

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