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# Around the World in 80 Questions (Day 3): If a and b are integers and

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Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink]
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Bunuel wrote:
If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ?

A. 14
B. 12
C. 10
D. 7
E. 6

 This question was provided by GMAT Club for the Around the World in 80 Questions Win over $20,000 in prizes: Courses, Tests & more The number 729 is 3^6. Therefore, the possible pairs of (a, b) are 7 pairs (1, 729), (3, 243), (9, 81), (27, 27), (81, 9), (243, 3), and (729, 1). We also have to consider the negative pairs so 7 more pairs. (-1, -729), (-3, -243), (-9, -81), (-27, -27), (-81, -9), (-243, -3), and (-729, -1). Therefore 7+7=14 Hence A: 14. Manager Joined: 07 May 2023 Posts: 198 Own Kudos [?]: 242 [0] Given Kudos: 47 Location: India Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] Bunuel wrote: If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ? A. 14 B. 12 C. 10 D. 7 E. 6  This question was provided by GMAT Club for the Around the World in 80 Questions Win over$20,000 in prizes: Courses, Tests & more

729 = 3^6

Hence, we can have 7 factors.

In an order pair, the value of a, and b matter. Hence (1,729) and (729,1) is considered different.

Number of order pairs = 7*2 = 14

IMO A
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Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink]
2
Kudos
given that a, b are integers and a*b = 3^6
positive pairs can be
3^6 *1
3^5 * 3
3^4*3^2
3^3*3^3
3^2*3^4
3^1 * 3^5
1*3^6

7 pairs

-ve pairs
(-3)^6*1
(-3)^5 * -3
(-3)^4* (-3)^2
(-3)^3 * ( -3)^3
(-3)^2 * (-3)^4
(-3)^1 * (-3)^5
(1* (-3)^6)

7 pairs

total 14 pairs possible

OPTION A

Bunuel wrote:
If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ?

A. 14
B. 12
C. 10
D. 7
E. 6

 This question was provided by GMAT Club for the Around the World in 80 Questions Win over $20,000 in prizes: Courses, Tests & more Manager Joined: 07 May 2023 Posts: 198 Own Kudos [?]: 242 [0] Given Kudos: 47 Location: India Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] Obscurus wrote: Bunuel wrote: If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ? A. 14 B. 12 C. 10 D. 7 E. 6  This question was provided by GMAT Club for the Around the World in 80 Questions Win over$20,000 in prizes: Courses, Tests & more

729 = 3^6

Hence, we can have 7 factors.

In an order pair, the value of a, and b matter. Hence (1,729) and (729,1) is considered different.

Number of order pairs = 7*2 = 14

IMO A

bb Bunuel

Checking if my response was evaluated. The answer is correct, however kudos wasn't awarded.

Thanks.

https://gmatclub.com/forum/around-the-w ... l#p3236467
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Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink]
Obscurus wrote:
Obscurus wrote:
Bunuel wrote:
If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ?

A. 14
B. 12
C. 10
D. 7
E. 6

 This question was provided by GMAT Club for the Around the World in 80 Questions Win over $20,000 in prizes: Courses, Tests & more 729 = 3^6 Hence, we can have 7 factors. In an order pair, the value of a, and b matter. Hence (1,729) and (729,1) is considered different. Number of order pairs = 7*2 = 14 IMO A bb Bunuel Checking if my response was evaluated. The answer is correct, however kudos wasn't awarded. Thanks. https://gmatclub.com/forum/around-the-w ... l#p3236467 Logic is incorrect. Hence no kudos. Manager Joined: 07 May 2023 Posts: 198 Own Kudos [?]: 242 [0] Given Kudos: 47 Location: India Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] [quote="Bunuel" Logic is incorrect. Hence no kudos.[/quote] Bunuel Thank you for the feedback. I looked into some of the responses which were indicated correct. I think you're referring to the negative factors. If so, I agree that I didn' t consider that. But if that's the case, isn't none of the option correct. The questions asks us to find the ordered pair. In an ordered pair (a,b) $$\neq$$ (b,a), that's why the term ordered pair is used. Otherwise the pair is un-ordered. Therefore the correct number of ordered pair should be 28 and not 14. For example (1,729), (729,1), (-1, -729) and (-729,-1) are all different ordered pairs. Can you please clarify this. Math Expert Joined: 02 Sep 2009 Posts: 93945 Own Kudos [?]: 633628 [1] Given Kudos: 82411 Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] 1 Kudos Expert Reply Obscurus wrote: Thank you for the feedback. I looked into some of the responses which were indicated correct. I think you're referring to the negative factors. If so, I agree that I didn' t consider that. But if that's the case, isn't none of the option correct. The questions asks us to find the ordered pair. In an ordered pair (a,b) $$\neq$$ (b,a). Therefore the correct number of ordered pair should be 28 and not 14. For example (1,729), (729,1), (-1, -729) and (-729,-1) are all different ordered pairs. Can you please clarify this. Sure, here are possible pairs: 1. a = -729 and b = -1 2. a = -243 and b = -3 3. a = -81 and b = -9 4. a = -27 and b = -27 5. a = -9 and b = -81 6. a = -3 and b = -243 7. a = -1 and b = -729 8. a = 1 and b = 729 9. a = 3 and b = 243 10. a = 9 and b = 81 11. a = 27 and b = 27 12. a = 81 and b = 9 13. a = 243 and b = 3 14. a = 729 and b = 1 Hope it helps. Manager Joined: 07 May 2023 Posts: 198 Own Kudos [?]: 242 [0] Given Kudos: 47 Location: India Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] Bunuel wrote: Obscurus wrote: Thank you for the feedback. I looked into some of the responses which were indicated correct. I think you're referring to the negative factors. If so, I agree that I didn' t consider that. But if that's the case, isn't none of the option correct. The questions asks us to find the ordered pair. In an ordered pair (a,b) $$\neq$$ (b,a). Therefore the correct number of ordered pair should be 28 and not 14. For example (1,729), (729,1), (-1, -729) and (-729,-1) are all different ordered pairs. Can you please clarify this. Sure, here are possible pairs: 1. a = -729 and b = -1 2. a = -243 and b = -3 3. a = -81 and b = -9 4. a = -27 and b = -27 5. a = -9 and b = -81 6. a = -3 and b = -243 7. a = -1 and b = -729 8. a = 1 and b = 729 9. a = 3 and b = 243 10. a = 9 and b = 81 11. a = 27 and b = 27 12. a = 81 and b = 9 13. a = 243 and b = 3 14. a = 729 and b = 1 Hope it helps. Thank you so much. I realized my mistake. Tutor Joined: 26 Jun 2014 Status:Mentor & Coach | GMAT Q51 | CAT 99.98 Posts: 450 Own Kudos [?]: 791 [0] Given Kudos: 8 Re: Around the World in 80 Questions (Day 3): If a and b are integers and [#permalink] Expert Reply Bunuel wrote: If a and b are integers and a*b = 729, how many ordered pairs of (a, b) are possible ? A. 14 B. 12 C. 10 D. 7 E. 6  This question was provided by GMAT Club for the Around the World in 80 Questions Win over$20,000 in prizes: Courses, Tests & more

Since 729 = 3^6, and we need to multiply a and b to get 3^6, we must ensure that a and b are of the form 3^p and 3^q where p and q are integers (note that -3^p and -3^q is also possible, but more of that later)

Thus, we have: 3^p * 3^q = 3^6
=> p + q = 6

Thus, there are 7 non-negative integer solutions: 0+6, 1+5, 2+4, 3+3, 4+2, 5+1, 6+0
Thus, the solutions here are 3^0 * 3^6, 3^1 * 3^5, etc.
Note: Number of non-negative integer solutions for the equation p + q = N is (N + 1)

Considering the negative solutions, there are 7 more solutions; for example: (-3^0) * (-3^6)

Thus, there are 7 + 7 = 14 solutions