Is the positive integer x divisible by each integer from 2 through 6 ?

considering the numbers one at a time in order to figure out what x will need, at a minimum, in it's prime factorization:

2: in order for x to be divisible by 2, x needs a 2 in its PF
3: in order for x to be divisible by 3, x needs a 3 in its PF (so now we have 2*3)
4: in order for x to be divisible by 4, x needs another 2 in its PF (so now we have 2*2*3)
5: in order for x to be divisible by 5, x needs a 5 in its PF (so now we have 2*2*3*5)
6: in order for x to be divisible by 6, x needs a 2 and a 3 in its PF (we already have that so our min doesn't change; x still needs, at min, 2*2*3*5 in its PF)

So we need enough info to prove that x has at least 2*2*3*5 in its PF

Statement 1.

x = (10)(m) ---> rewriting m in terms of what we know about its PF --> x=(2*5)(2*3*2*5*whatever else might be in m)

We can see right away that x has at least 2*2*3*5 in its PF. Sufficient.

Statement 2

10x = n ---> x = n/10 --> rewriting m in terms of what we know about its PF, and factoring 10 -->
x = (2*2*2*3*3*5*7*whatever else might be in n)/2*5 --> the two and 5 cancel to get --->
x = (2*2*3*3*7*whatever else might be in n)

We don't know if x has 2*2*3*5 in its PF because we don't know for sure that there is a 5; there may or may not be a 5 in the 'whatever else might be in n' part. Insufficient.

A is the answer.

If we modify the original condition and the question, since x has to be divisible by 2, 3, 4, 5 and 6, the question becomes “is it x a multiple of 60?”
In case of con 1), from x=10m, m is divisible by 2, 3, 4 and 5. Hence, x is a multiple of 60. The answer is yes and the condition is sufficient. Thus, the correct answer is A.

- Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Hello, can someone please explain to me why can we add another five in the second statement?
It's kind of driving me nuts!

Hi,

St2: 10x = n --> x = n/10
n can be any integer.
If n = 2*3*4*5*6*7*8*9*5, then x is divisible by each integer from 2 through 6 --> Another 5 is added here to show sufficiency.
If n = 2*3*4*5*6*7*8*9, then x is not divisible by 5 because n/10 consumes a 2 and 5 --> In this case St2 is not sufficient
We can conclude that St2 is not sufficient at all times.

Hope it helps

Bunuel

I got this question in GMATPREP 5 exam. Please tag this to exampack 2

_________________
Done. Thank you.

I have a simpler approach

m is div by 2 to 5
Write all primes then just see what's missing and add it into the list

SEE
m = 2x3x5 x(2) is the smallest div by all no.s 2-5

Similarly in the second one

Excellent explanation brunel.

We need to check whether x is divisible by 4,5,6 (if yes, then definitely x is divisible by 2 and 3 as well)

St 1: x = 2 * 5 * m
Since "m is a positive integer divisible by each integer from 2 through 5"; then m is divisible by 6 as well. If m is divisible by 4,5, 6, then x is also divisible by 4,5,6.
Hence St 1 is sufficient by iself

St 2: x = n/10; we know that n is divisible by 4,5,6 but we cannot say for sure that x is also divisible by 5. Hence not sufficient.

Option A is the correct answer.

I got a little confused with the wording of the question...

because 10m can be interpreted as (10)(m)=10m,
or it can mean 10m where m is a positive integer divisible by each integer from 2 through 5 -> 102, 103, 104, 105

No. 10m can only mean 10*m. If it were a three digit number, then it would be explicitly mentioned.

Is the positive integer x divisible by each integer from 2 through 6 ?

(1) x = 10m, where m is a positive integer divisible by each integer from 2 through 5

We need to check if x is divisible by 2,3,4,5,6 or not.
Its given that m is divisible by 2,3,4,5. That means x is also divisible by 2,3,4,5.
So now, we are left with the divisibility of 6. Since x is divisible by 2 and 3, we can confirm that x will be also be divisible by 6.

So Statement 1 alone is sufficient.


(2) 10x = n, where n is a positive integer divisible by each integer from 2 through 9

Its given that , n is divisible by 2,3,4,5,6,7,8,9.

10x = n
2 * 5 * x = n

Clearly we can say that x will be divisible by 3,7,9 and for rest of the numbers, we need to analyze more.
Since n is divisible by 4, then x should be divisible by 2, as there is only one factor of 2 in 10.

Also we cannot confirm that x is divisible by 5 as 5 is a factor in 10.
Example:
Its given that 10x is divisible by 5
if x= 1, 10x = 10 * 1 = 10 is divisible by 5 but x is not divisible by 5.
if x =10 , 10x = 10*10 = 100 is divisible by 5. But here x is divisible by 5.
So we cannot confirm that X is divisible by 5 in this case. Both cases are possible.

Similarly n is divisible by 8, then x should be divisible by 4, as there is only one factor of 2 in 10.
Also x is divisible by 2 and 3, then it should be divisible by 6 as well .

So here we can confirm that x is divisible by 2,3,4,6. But we are not sure if its divisible by 5 or not. Hence Insufficient.

Option A is the answer.

Thanks,
Clifin J Francis
GMAT SME

Simply, the first condition tells you you have all of the factors in the numbers from 2 to 5, which would also happen to include 6 as a factor because you have a 2 and a 3 among the factors already. In the second condition, when you solve for x by dividing out the 10, the 5 will cancel and you don't have 5 as a factor anymore. Not sufficient

I did it this way.

1. Since x = 10n, we can already see this is at least divisible by 2 and 5. Now, based on what the statement 1 says, ix is also divisible by 3,4. Hence, x is also divisible by 2x3 = 6. SUFFICIENT
2. Here, 10x = n. For statement 2 I tested certain cases. Since n is divisible by 2 through 9, we cannot say x is divisible by all these numbers. For example, 10 is divisible by 2 and 5. If x = 60 then the entire number is divisible by 2 and 5 but if x is 54, then it is divisible by 9 and 6 but not 2 and 5. INSUFFICIENT

Hence, (A) is the answer.

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