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17 Oct 2009, 18:33
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New questions:
1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/abcde-is-a-r ... 86284.html
2. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) =
(A) P-1
(B) P-2
(C) (P+1)/2
(D) (P-1)/2
(E) 2
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/the-function ... 03852.html
3. How many numbers that are not divisible by 6 divide evenly into 264,600?
(A) 9
(B) 36
(C) 51
(D) 63
(E) 72
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-num ... 26647.html
4.A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?
(A) 20
(B) 36
(C) 48
(D) 60
(E) 84
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/a-certain-qu ... 96389.html
5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?
(A) 13
(B) 14
(C) 15
(D) 16
(E) more than 16
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/mrs-smith-ha ... 98225.html
6. This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?
(A) 1/(r+2)
(B) 1/(2r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/this-year-he ... 00891.html
7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I?
(A) 50+I/200
(B) 50+3I/100
(C) 50+I/40
(D) 100+I/50
(E) 100+3I/100
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/before-being ... 25936.html
8. How many positive integers less than 10,000 are such that the product of their digits is 210?
(A) 24
(B) 30
(C) 48
(D) 54
(E) 72
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-pos ... 02927.html
9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE".
(A) 70
(B) 36
(C) 35
(D) 72
(E) 32
Find in the above word, the number of arrangements using the 4 letters.
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/find-the-num ... 98237.html
10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1≤x≤3 and 1≤y≤3?
(A) 72
(B) 76
(C) 78
(D) 80
(E) 84
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-tri ... 98236.html
Also you can check new set of DS problems: https://gmatclub.com/forum/new-set-of-good-ds-85441.html
1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/abcde-is-a-r ... 86284.html
2. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) =
(A) P-1
(B) P-2
(C) (P+1)/2
(D) (P-1)/2
(E) 2
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/the-function ... 03852.html
3. How many numbers that are not divisible by 6 divide evenly into 264,600?
(A) 9
(B) 36
(C) 51
(D) 63
(E) 72
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-num ... 26647.html
4.A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?
(A) 20
(B) 36
(C) 48
(D) 60
(E) 84
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/a-certain-qu ... 96389.html
5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?
(A) 13
(B) 14
(C) 15
(D) 16
(E) more than 16
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/mrs-smith-ha ... 98225.html
6. This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?
(A) 1/(r+2)
(B) 1/(2r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/this-year-he ... 00891.html
7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I?
(A) 50+I/200
(B) 50+3I/100
(C) 50+I/40
(D) 100+I/50
(E) 100+3I/100
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/before-being ... 25936.html
8. How many positive integers less than 10,000 are such that the product of their digits is 210?
(A) 24
(B) 30
(C) 48
(D) 54
(E) 72
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-pos ... 02927.html
9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE".
(A) 70
(B) 36
(C) 35
(D) 72
(E) 32
Find in the above word, the number of arrangements using the 4 letters.
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/find-the-num ... 98237.html
10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1≤x≤3 and 1≤y≤3?
(A) 72
(B) 76
(C) 78
(D) 80
(E) 84
OPEN DISCUSSION OF HIS QUESTION IS HERE: https://gmatclub.com/forum/how-many-tri ... 98236.html
Also you can check new set of DS problems: https://gmatclub.com/forum/new-set-of-good-ds-85441.html
23 Oct 2009, 23:42
Kudos
Bookmarks
5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?
(A) 13
(B) 14
(C) 15
(D) 16
(E) more than 16
Answer: C.
Clearly there are more than 8 vouchers as each of four can get at least 2. So, basically 120 ways vouchers can the distributed are the ways to distribute \(x-8\) vouchers, so that each can get from zero to \(x-8\) as at "least 2", or 2*4=8, we already booked. Let \(x-8\) be \(k\).
In how many ways we can distribute \(k\) identical things among 4 persons? Well there is a formula for this but it's better to understand the concept.
Let \(k=5\). And imagine we want to distribute 5 vouchers among 4 persons and each can get from zero to 5, (no restrictions).
Consider:
\(ttttt|||\)
We have 5 tickets (t) and 3 separators between them, to indicate who will get the tickets:
\(ttttt|||\)
Means that first nephew will get all the tickets,
\(|t|ttt|t\)
Means that first got 0, second 1, third 3, and fourth 1
And so on.
How many permutations (arrangements) of these symbols are possible? Total of 8 symbols (5+3=8), out of which 5 \(t\)'s and 3 \(|\)'s are identical, so \(\frac{8!}{5!3!}=56\). Basically it's the number of ways we can pick 3 separators out of 5+3=8: \(8C3\).
So, # of ways to distribute 5 tickets among 4 people is \((5+4-1)C(4-1)=8C3\).
For \(k\) it will be the same: # of ways to distribute \(k\) tickets among 4 persons (so that each can get from zero to \(k\)) would be \((K+4-1)C(4-1)=(k+3)C3=\frac{(k+3)!}{k!3!}=120\).
\((k+1)(k+2)(k+3)=3!*120=720\). --> \(k=7\). Plus the 8 tickets we booked earlier: \(x=k+8=7+8=15\).
Answer: C (15).
P.S. Direct formula:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is \(n+r-1C_{r-1}\).
The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).
Hope it helps.
(A) 13
(B) 14
(C) 15
(D) 16
(E) more than 16
Answer: C.
Clearly there are more than 8 vouchers as each of four can get at least 2. So, basically 120 ways vouchers can the distributed are the ways to distribute \(x-8\) vouchers, so that each can get from zero to \(x-8\) as at "least 2", or 2*4=8, we already booked. Let \(x-8\) be \(k\).
In how many ways we can distribute \(k\) identical things among 4 persons? Well there is a formula for this but it's better to understand the concept.
Let \(k=5\). And imagine we want to distribute 5 vouchers among 4 persons and each can get from zero to 5, (no restrictions).
Consider:
\(ttttt|||\)
We have 5 tickets (t) and 3 separators between them, to indicate who will get the tickets:
\(ttttt|||\)
Means that first nephew will get all the tickets,
\(|t|ttt|t\)
Means that first got 0, second 1, third 3, and fourth 1
And so on.
How many permutations (arrangements) of these symbols are possible? Total of 8 symbols (5+3=8), out of which 5 \(t\)'s and 3 \(|\)'s are identical, so \(\frac{8!}{5!3!}=56\). Basically it's the number of ways we can pick 3 separators out of 5+3=8: \(8C3\).
So, # of ways to distribute 5 tickets among 4 people is \((5+4-1)C(4-1)=8C3\).
For \(k\) it will be the same: # of ways to distribute \(k\) tickets among 4 persons (so that each can get from zero to \(k\)) would be \((K+4-1)C(4-1)=(k+3)C3=\frac{(k+3)!}{k!3!}=120\).
\((k+1)(k+2)(k+3)=3!*120=720\). --> \(k=7\). Plus the 8 tickets we booked earlier: \(x=k+8=7+8=15\).
Answer: C (15).
P.S. Direct formula:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is \(n+r-1C_{r-1}\).
The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is \(n-1C_{r-1}\).
Hope it helps.
23 Oct 2009, 23:00
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Bookmarks
Economist
3. How many numbers that are not divisible by 6 divide evenly into 264,600?
(A) 9
(B) 36
(C) 51
(D) 63
(E) 72
Answer: D.
First of all you should know the formula counting the number of distinc factors of an integer:
You have to write the number as the product of primes as a^p*b^q*c^r, where a, b, and c are prime factors and p,q, and r are their powers.
The number of factors the number contains will be expressed by the formula (p+1)(q+1)(r+1).
Let's take an example for clear understanding:Find the number of all (distinct) factors of 1435:
1. 1435 can be expressed as 5^1*17^1*19^1
2. total number of factors of 1435 including 1 and 1435 itself is (1+1)*(1+1)*(1+1)=2*2*2=8 factors.
OR
Distinct factors of 18=2*3^2 --> (1+1)*(2+1)=6. Lets check: factors of 18 are: 1, 2, 3, 6, 9 an 18 itself. Total 6.
Back to our question:
How many numbers that are not divisible by 6 divide evenly into 264,600?
264,600=2^3*3^3*5^2*7^2
We should find the factor which contain no 2 and 3 together, so not to be divisible by 6.
Clearly, the factors which contain only 2,5,7 and 3,5,7 won't be divisible by 6. So how many such factors are there?
2^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36 (the product of powers of 2, 5,and 7 added 1)
3^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36 (the product of powers of 2, 5, and 7 added 1)
So 36+36=72. BUT this number contains duplicates:
For example: 2^3*5^2*7^2--> (3+1)*(2+1)*(2+1)=36 This 36 contains the factors when the power of 2 is 0 (2^0=1)--> 2^0*5^2*7^2 giving us only the factors which contain 5-s and/or 7-s. (5*7=35, 5*7^2=245, 5^2*7=175, 5*7^0=5, 5^0*7=7....) number of such factors are (2+1)*(2+1)=9 (the product of powers of 5 and 7 added 1).
And the same factors are counted in formula 3^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36: when power of 3 is 0 (3^0=1). --> 5*7=35, 5*7^2=245, 5^2*7=175, 5*7^0=5, 5^0*7=7.... such factors are (2+1)*(2+1)=9. (the product of powers of 5 and 7 added 1).
So we should subtract this 9 duplicated factors from 72 --> 72-9=63. Is the correct answer.
The problem can be solved from another side:
264,600=2^3*3^3*5^2*7^2 # of factors= (3+1)(3+1)(2+1)(2+1)=144. So our number contains 144 distinct factors. # of factors which contain 2 and 3 is 3*3=9 (2*3, 2^2*3, 2^3*3, 2*3^2, 2^2*3^2, 2^3*3^2, 2*3^3, 2^2*3^3, 2^3*3^3 total 9) multiplied by (2+1)*(2+1)=9 (powers of 5 and 7 plus 1) --> 9*9=81 ---> 144-81=63.
Hope now it's clear.
