N is the smallest number which is divisible by 49 and has all its digi

Question

N is the smallest number which is divisible by 49 and has all its digit same. How many prime numbers are factors of N ?

a) 3
b) 9
c) 4
d) 7
e) 5

mikemcgarry · Answer

Dear manpreetsingh86 I'm happy to respond. :-)

For a few reasons, this is not really a GMAT question. First of all, on the GMAT, the answers choices are always always listed in strict numerical order, and these choices are not. Second, the wording is not precise. This number has six prime factors (not an answer) but two of them are the same, so there are five distinct prime factors. The text of the question omitted that crucial word. Finally, I do a LOT of math in my head, but I needed a calculator for this. It's a little beyond me how anyone other than a savant would do this without a calculator. As I found with aid of a calculator 777,777 = 49*15,873 = (3)(7^2)(11)(13)(37) Those are my thoughts. Mike :-)

manpreetsingh86 · Answer

hi mike, thanks for responding and clearing the air regarding the question. i got this question, from one of my friend, who is also preparing for gmat at the well known institute over here in india.

well, regarding the solution. this is how, i responded to this question.

Rule : All numbers whose digits are same and are in a multiple of six will always be divisible by 7,11 and 13. thus 111111,111111111111 will always be divisible by 7,11 and 13.

so, we can quickly zero down to number 111111. we can see this number is not divisible by 49. thus 777777 is our lucky number. next, by using the rule mentioned above, without doing any calculation we can see that it will be divisible by 11 and 13. also since, sum of its digits is a multiple of 3. therefore, it will be divisible by 3. thus, without touching any calculator we can zero down to 4 distinct prime factors. now as far as the fifth prime factor is concerned, we can focus on the unit digit( which is 7) of original number.
now, unit of 7^2=9, 13=3,3=3 and 11=1 
thus unit digit of product of 3,7^2,11 and 13 is 1, thus our next prime must have a unit digit of 7. thus our first two possible choices are 17 and 37. we will first divide the number by 17. we will see, its not divisible by 17. next we will divide the number by 37 and bingo.

As can be see, until the last step,we are not required to do any calculation.

P.S. : this is an unsolved question. i've pinged the link of the same question to the mike for his consideration.

KarishmaB · Answer

If this is the kind of questions the institute is passing off as GMAT questions, you really need to worry whether you (or your friend) are in the right hands. As a test maker, I can tell you that it is very easy to make hard questions. The problem lies in making simple questions which seem hard - the requirement of GMAT. The problem lies in making them GMAT relevant!

There is no logical way forward in this question. N is a multiple of 49 with all its digits same. There are innumerable numbers with all digits same - 22, 222, 2222, 3, 33, 333, 3333, 33333, 33333333 etc.

One cannot pick up a random rule and take that as starting point. We encourage people to understand logic, not learn up rules. Here, the question requires one to learn up obscure rules. Even if one does know that 6 digit numbers with all same digits will be divisible by 7, how do you know that a number such as 9999 is not divisible by 49 without calculating?

Mind you, rules can come in handy. In fact, I will give you a question I made for which this rule is relevant. But knowing the rule only hastens your solving, not knowing is not an impediment. Try this:

Question: On multiplying a positive integer N having n digits by (n + 2), we get a number with (n + 1) digits, all of whose digits are (n + 1). How many such N exist?
(A) None
(B) 1
(C) 2
(D) 8
(E) 9

There is a logical starting point here which gives you a range of only a few numbers. You ignore some of them using logic. You work on only a few and find your answer.

manpreetsingh86 · Answer

hi karishma, thank you very much for responding. well i'm not taking any coaching from any institute and i'm not associated with any coaching institute. usually, when we prepare for something, we tend to share our knowledge about the subject with other people. my friend has got this question, from his faculty. he put the same question to me. so, i thought of discussing the same here in this forum. i don't see any harm in doing such thing. Also, because i'm not an expert, so therefore don't know which question qualifies as gmat type and which doesn't.

well, this is definitely not some random rule. i know about these stuff and other rules, back from my olympiad days. i thought others might be aware of it, if not, then they might learn something new today.

And the answer to your question is 9.

manpreetsingh86 · Answer

well, here is another method to solve this question.

as all digits of the number are same. so let's start from a three digit number. 111 is not divisible by 7. since 111 is not divisible by 7, thus any three digit number having all of its digit same, except 777 will not divisible by 7. why ? because all these number will be multiple of 111 and 111 is not a factor of 7.

Also, 777=7(111). as 111 is not divisible by 7. thus 777 will not be divisible by 49.

by using the similar reasoning we can come straight to number 111111. which is divisible by 7, but not 49. thus 777777 will be our magic number.
clearly, this number is divisible by 7,3 (as sum of its digits is divisible by 3) and 11 (111111 can be written as 11,11,11). now,check the unit digit of the number obtained so far.

which is 9*3*1=7. thus we need those prime number which results in 1 in the unit digit when multiplied.(13,17,23,37 can be such numbers) let's try 13 gotcha. thus we only have to find the prime number which has 7 at its unit place. thus we will first try number 17, clearly not divisible. then we will divide the number by 37 and bingo.

as can be seen, this question can also be solved by not getting affected with savant and by not remembering any random rule.

KarishmaB · Answer

The question is barely suitable for CAT, no matter how you do it, let alone GMAT. It is common in India to pass off the CAT questions as GMAT questions (I have worked in an institute in India so I know) since GMAT is and will remain a sidekick there - it doesn't provide enough ROI. So if you choose to ignore our advice, that's certainly your call, but in my opinion, it is worthwhile to invest your time in only GMAT specific questions if you are going to take only GMAT.

KarishmaB · Answer

As for this question, here is the solution:

Let’s take an example to understand the question better. 
"On multiplying a positive integer N having n digits by (n + 2), we get a number with (n + 1) digits, all of whose digits are (n + 1)"

Example: Let n = 3

_ _ _ × 5 = 4444

The numbers on the right hand side of the equation can only be of the form 1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999 which gives us a limited set of possible options. 
Looking at the example, we can find the 3 digit number, N, by dividing 4444 by 5. But since 4444 is not divisible by 5, there is no such positive integer.
Similarly, all we have to do is, using divisibility rules, divide each of these 9 numbers and see if they are divisible by their respective (n + 2)s. 1 is not divisible by 2, 22 is not divisible by 3, 333 is not divisible by 4 and so on… An odd number will never be divisible by an even number so we don’t need to check for odd numbers. So we ignore 1, 33, 55555, 7777777, 999999999

We find only 666666 is divisible by 7 giving 95338 as N. So there is only one such N.

Answer (B)

This question is at Q50/51, even though the starting point is clear, there is a small set of values and calculations involved are almost zero.

manpreetsingh86 · Answer

hi karishma, thanks for your reply, i made a careless mistake here. i assume that all digits of the new number are same. so end up getting 9 (from 111111 to 999999).

I'm not arguing regarding the quality of the question. i already thanked mike for his suggestion on the quality of this question. i was just trying to put forward the point that this question can be solved with more logic and less math. In my first post, i used one of the rule to solve the question and in the second solution, i only used logic nothing else.

i again, thank you karishma for being part of this discussion and giving your expert opinion, it indeed was helpful.

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N is the smallest number which is divisible by 49 and has all its digi

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