How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 amFollowed by: 2 members

by Gmat_mission » Sat Aug 22, 2020 4:06 am

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How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

A. 111
B. 112
C. 333
D. 334
E. 1,134

Answer: B

Source: Official Guide

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Source: Beat The GMAT — Problem Solving | Sat Aug 22, 2020 4:06 am

priyasinglaa: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Sat Aug 22, 2020 9:41 amFollowed by: 9 members

Re: How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

by priyasinglaa » Sat Aug 22, 2020 10:58 pm

1000 < =3n <= 9999
o 1000/3 <= n < = 9999/3
o 333.33 < = n < 3333-----(1)
• 1000 < =n/3 <= 9999
o 3000 < = n < 9999*3---------(2)

• Combining both 1 and 2, we get
o 3000 <= n < 3333
o As n/3 is an integer, n has to be a multiple of 3.
o (3333-3000)/3 +1 = 112
Hence, n can have 112 values.

Thus, option B is the correct answer.

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Aman Faruki: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Mon Aug 24, 2020 6:50 am

Re: How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

by Aman Faruki » Mon Aug 24, 2020 7:01 am

Thanks Priya!

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Rinksss: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Mon Aug 24, 2020 7:16 am

Re: How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

by Rinksss » Mon Aug 24, 2020 7:26 am

Thanks

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Reenadddd: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Mon Aug 24, 2020 7:02 am

Re: How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}3$ are 4-digit integers?

by Reenadddd » Mon Aug 24, 2020 7:29 am

Thanks

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