Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized for You
we will pick new questions that match your level based on your Timer History
Track Your Progress
every week, we’ll send you an estimated GRE score based on your performance
Practice Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Your score will improve and your results will be more realistic
Is there something wrong with our timer?Let us know!
m is a three-digit integer such that when it is divided by 5, the remainder is y, and when it is divided by 7, the remainder is also y. If y is a positive integer, what is the smallest possible value of m?
Re: m is a three-digit integer such that when it is divided by
[#permalink]
12 Aug 2017, 13:44
6
6
Bookmarks
Carcass wrote:
m is a three-digit integer such that when it is divided by 5, the remainder is y, and when it is divided by 7, the remainder is also y. If y is a positive integer, what is the smallest possible value of m?
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
When m is divided by 5, the remainder is y So, m = 5k + y for some integer k
When m is divided by 7, the remainder is y So, m = 7j + y for some integer j
Since both equations are set to equal m, we can write: 5k + y = 7j + y Subtract y from both sides to get: 5k = 7j Well, 5k represents a multiple of 5, and 7j represents a multiple of 7 So, what's the smallest 3-digit number that is a multiple of 5 AND a multiple of 7?
The smallest 3-digit number is 100 100 is a multiple of 5, but it's NOT a multiple of 7
Next we have 105 105 is a multiple of 5, AND it's a multiple of 7 Now be careful. This does NOT mean that m = 105
When we divide 105 by 5 we get a remainder of 0, but we're told that the remainder (y) is a POSITIVE INTEGER. To MINIMIZE the value of m, we need a super small remainder. The smallest possible non-zero remainder is 1. 105 + 1 = 106
So, 106 is the smallest possible 3-digit value of m.
Re: m is a three-digit integer such that when it is divided by
[#permalink]
07 Jul 2018, 12:25
1
GreenlightTestPrep wrote:
Carcass wrote:
m is a three-digit integer such that when it is divided by 5, the remainder is y, and when it is divided by 7, the remainder is also y. If y is a positive integer, what is the smallest possible value of m?
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
When m is divided by 5, the remainder is y So, m = 5k + y for some integer k
When m is divided by 7, the remainder is y So, m = 7j + y for some integer j
Since both equations are set to equal m, we can write: 5k + y = 7j + y Subtract y from both sides to get: 5k = 7j Well, 5k represents a multiple of 5, and 7j represents a multiple of 7 So, what's the smallest 3-digit number that is a multiple of 5 AND a multiple of 7?
The smallest 3-digit number is 100 100 is a multiple of 5, but it's NOT a multiple of 7
Next we have 105 105 is a multiple of 5, AND it's a multiple of 7 Now be careful. This does NOT mean that m = 105
When we divide 105 by 5 we get a remainder of 0, but we're told that the remainder (y) is a POSITIVE INTEGER. To MINIMIZE the value of m, we need a super small remainder. The smallest possible non-zero remainder is 1. 105 + 1 = 106
So, 106 is the smallest possible 3-digit value of m.
RELATED VIDEO
How could i know which numbers i should use in general? What is your strategy of restricting the range of possible values for this problem?
Re: m is a three-digit integer such that when it is divided by
[#permalink]
07 Jul 2018, 13:24
3
Expert Reply
This question is quite tricky.
However, 99% of the time, the more a tricky question is, the more there is always a shortcut or at a closer inspection a solution is suddenly behind the curve.
Reading carefully the stem, it says that you have to consider a 3 digit integer number and you have to find the least possible value.
So, a 3 digit number at least is 100. Now, you do also know that when it is divided by 5 y is the remainder and when it is divided by 7 is yet y the remainder, which means is the same number.
A common number that divides evenly 7 and 5 is 105. 105 divided by 5 AND 7 has no rest or reminder. From this is easy to think that a number divided by both 5 and 7 with the same reminder y, for instance, the reminder is 1, is 106.
106 is the least number with 3 digits you can have when you divide it by 5 and 7.
Re: m is a three-digit integer such that when it is divided by
[#permalink]
26 Jul 2021, 11:30
The way to solve this problem is to realize that if you solve for the least common multiple of 5 and 7 that has 3 digits, it leaves you with 105. If you divide 105 by 5 and 7 your remainder is 0. If you add one to 105 = 106, then your remainder is 1 for both values.
Re: m is a three-digit integer such that when it is divided by
[#permalink]
20 May 2022, 20:56
Simple: Three digit numbers start from 100, and go as 100, 101, 102, 103, 104, 105, 106,....
Remember that the remainder should always be lesser than divisor. Thus y should be lesser than 5 and 7. Note that Smallest possible positive remainder is always 1. Thus look for numbers in the above sequence that when divided by 5 and 7 yield remainder 1. Thus 106 is the answer.
Re: m is a three-digit integer such that when it is divided by
[#permalink]
30 Jul 2024, 00:39
Hello from the GRE Prep Club BumpBot!
Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
gmatclubot
Re: m is a three-digit integer such that when it is divided by [#permalink]