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Math Expert V
Joined: 02 Sep 2009
Posts: 65014
How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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12 00:00

Difficulty:   95% (hard)

Question Stats: 32% (02:48) correct 68% (02:54) wrong based on 47 sessions

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How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

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Math Expert V
Joined: 02 Aug 2009
Posts: 8753
Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

So y = ax where a= 1, 2, 3....

We will now calculate for each value of ax..
1) $$a=1...2x + x < 40....3x<39+1....x<13+\frac{1}{3}.......1\leq{x}\leq{13}....13$$ values
2) $$a=2...2x + 2x < 40....4x<40....x<10.......1\leq{x}<{10}....9$$ values
3) $$a=3...2x + 3x < 40....5x<40....x<8.......1\leq{x}<8....7$$ values
4) $$a=4...2x + 4x < 40....6x<36+4....x<6+\frac{4}{6}.......1\leq{x}\leq{6}....6$$ values
5) $$a=5...2x + 5x < 40....7x<35+5....x<5+\frac{5}{7}.......1\leq{x}\leq{5}....5$$ values
6) Now 40/8 and 40/9 are between 4 and 5, so when a=6 and a=7, we will get x as 1 to 4, so 2*4=8 values
7) Now 40/10 to 40/13 are between 3 and 4, so when a=8 to 11, we will get x as 1 to 3, so 4*3=12 values
8) Now 40/14 and 40/19 are between 2 and 3, so when a=12 to 17, we will get x as 1 and 2, so 6*2=12 values
9) Now 40/20 to 40/39 are between 1 and 2, so when a=18 to 37, we will get x as 1, so 20*1=20 values..

Total = 13+9+7+6+5+8+12+12+20=92

C
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Director  P
Joined: 14 Dec 2019
Posts: 673
Location: Poland
GMAT 1: 570 Q41 V27
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Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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1
1
Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

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2x + y < 40 and y is a multiple of x.

1) When x = 1 => y< 38 => 37 values
2) When x = 2, y < 36 => 17 values (only multiples of 2)
3) When x = 3, y < 34 => 11 values (only multiples of 3)
4) When x = 4, y < 32 => 7 values (only multiples of 4)
5) When x = 5, y < 30 => 5 values (only multiples of 5)
6) When x = 6, y < 28 => 4 values (only multiples of 6)
7) When x = 7, y < 26 => 3 values (only multiples of 7)
8) When x = 8, y < 24 => 2 values (only multiples of 8)
9) When x = 9, y < 22 => 2 values (only multiples of 9)
10) When x = 10, y < 20 => 1 values (only multiples of 10)
11) When x = 11, y < 18 => 1 values (only multiples of 11)
12) When x = 12, y < 16 => 1 values (only multiples of 12)
13) When x = 13, y < 14 => 1 value (only 13)
13) When x = 14, y < 12 => 0 values

Total = 37+17+11+7+5+4+3+2+2+1+1+1+1 = 92

GMAT Club Legend  V
Status: GMATINSIGHT Tutor
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Posts: 4322
Location: India
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Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

Project PS Butler

Are You Up For the Challenge: 700 Level Questions

It's much harder and lengthy than expected questioning GMAT so I can only call it a fine practice to do just to clear concepts

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Joined: 14 Mar 2018
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Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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Bunuel - Please can we have we see your method of cracking this. This is one of the questions that might take more time than usual, hence approach needs to be crisp.
Manager  B
Joined: 17 Sep 2019
Posts: 60
Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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Can we solve this graphically?
Intern  B
Joined: 24 Nov 2019
Posts: 35
Location: India
WE: Analyst (Insurance)
Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

Project PS Butler

Are You Up For the Challenge: 700 Level Questions

Is there a shorter solution Bunuel?
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 11042
Location: United States (CA)
Re: How many pairs positive integers x and y satisfy 2x + y < 40, if y is  [#permalink]

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1
Bunuel wrote:
How many pairs positive integers x and y satisfy 2x + y < 40, if y is a multiple of x?

A. 90
B. 91
C. 92
D. 93
E. 94

Since x and y are positive integers and y is a multiple of x, then y ≥ x.

If x = 1, we have:

2(1) + y < 40

y < 38

So y can be any integer from 1 to 37, inclusive. Therefore, there are 37 pairs of positive integers x and y (or solutions) when x = 1.

If x = 2, we have:

2(2) + y < 40

y < 36

So y can be any even integer from 2 to 34, inclusive. Therefore, there are 17 solutions when x = 2.

If x = 3, we have:

2(3) + y < 40

y < 34

So y can be any multiple of 3 from 3 to 33, inclusive. Therefore, there are 11 solutions when x = 3.

If x = 4, we have:

2(4) + y < 40

y < 32

So y can be any multiples of 4 from 4 to 28, inclusive. Therefore, there are 7 solutions when x = 4.

If x = 5, we have:

2(5) + y < 40

y < 30

So y can be any multiples of 5 from 5 to 25, inclusive. Therefore, there are 5 solutions when x = 5.

If x = 6, we have:

2(6) + y < 40

y < 28

So y can be any multiples of 6 from 6 to 24, inclusive. Therefore, there are 4 solutions when x = 6.

If x = 7, we have:

2(7) + y < 40

y < 26

So y can be any multiples of 7 from 7 to 21, inclusive. Therefore, there are 3 solutions when x = 7.

If x = 8, we have:

2(8) + y < 40

y < 24

So y can be any multiples of 8 from 8 to 16, inclusive. Therefore, there are 2 solutions when x = 8.

If x = 9, we have:

2(9) + y < 40

y < 22

So y can be any multiples of 9 from 9 to 18, inclusive. Therefore, there are 2 solutions when x = 9.

If x = 10, we have:

2(10) + y < 40

y < 20

So y can only be 10. Therefore, there is only 1 solution when x = 10.

Similarly, when x = 11, 12 or 13, there is 1 solution for each of these x-values. If x ≥ 14, then there are no solutions. (For example, if x = 14, 2(14) + y < 40 → y < 12. However, recall that y ≥ x since y is a multiple of x.)

Therefore, the total number of solutions is:

37 + 17 + 11 + 7 + 5 + 4 + 3 + 2 x 2 + 1 x 4 = 54 + 18 + 12 + 4 + 4 = 72 + 20 = 92

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# How many pairs positive integers x and y satisfy 2x + y < 40, if y is   