Is the positive integer X divisible by 21?

I think, answer is C
statement 1 may be yes or no.
statement 2 may be yes or no.
But combining two statements you can see that from statement 1, X is divisible by 7 and from statement 2, X is divisible by 3. So X wiil be divisible by 21.

................
thanks

Nice question here the logic to see that in order to check the divisibility by 21 => number should be both divisible by 7 and 3
and we can conclude the D is the answer number would never be divisible by 21

oh! Now i understand.
thanks

Target question: Is the positive integer x divisible by 21?

Statement 1: When X is divided by 14, the remainder is 4
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

In this statement, we aren't told how many times 14 divides into x, but we can just assume that 14 divides into x k times. In other words, x divided by 14 equals k with remainder 4"
So, we can write: x = 14k + 4 (where k is some positive integer)
This is enough information to show that x is definitely NOT divisible by 21
How do I know this?

Take x = 14k + 4 and rewrite 14 as follows: x = (7)(2)k + 4
Highlight two values: x = (7)(2)k + 4
Simplify: x = (7)(some integer) + 4
This tells us that x is 4 GREATER than some multiple of 7
In other words, x is NOT divisible by 7.
If x is NOT divisible by 7, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When X is divided by 15, the remainder is 5
Let's say that 15 divides into x j times. In other words, x divided by 15 equals j with remainder 5"
So, we can write: x = 15j + 5 (where j is some positive integer)
Rewrite the equation: x = (3)(5)j + 5
Highlight two values: x = (3)(5)k + 5
Simplify: x = (3)(some integer) + 5
This tells us that x is 5 GREATER than some multiple of 3
In other words, x is NOT divisible by 3.
If x is NOT divisible by 3, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

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Here,
Statement 1 says:
x=18,32,46,60,74 and so on like that
So, the answer is No----->Sufficient
Statement 2 says:
x=20,35,50,65,80 and so on like that
So, the answer is NO---->Sufficient
So, the correct choice is
Thank you...

Good Question! Takes just above 2 minutes

(1) when x is divided by 14, the remainder is 4
--> x = 14A + 4, for any integer A
Possible values of x = 4, 18, 32, 46, 60, 74, 88
Remainder when divided by 21 = {4, 18, 11, 4, 18, 11, 4 . . . .REPEAT}
None of the possible values of x are divisible by 21 - Sufficient (A definite NO)

(2) when x is divided by 15, the remainder is 5
--> x = 15B + 5, for any integer B
Possible values of x = 5, 20, 35, 50, 65, 80, 95, 110, 125, 140 . . .
Remainder when divided by 21 = {5, 20, 14, 8, 2, 17, 11, 5, 20 ... REPEAT} -
None of the possible values of x are divisible by 21 - Sufficient (A definite NO)

IMO Option D

For x to be divisible by 21, x must have at least one 3 and one 7 in its factors

From statement 1 we can make out that x is even but not divisible by 14. This means that x has a 2 in its factor but not a 7

If x does not have a 7 in its factors, x is not divisible by 21

1 is sufficient

From statement 2 we can see that x is divisible by 5 but not 15. So this says that x does not have a 3 in its factors

If x does not have a 3 in its factors, x is not divisible by 21

2 is sufficient

Therefore Answer is (D)

Do we really need to go into this much depth when thinking about the problem?

Surely just by looking at X = 14q + 4 you can see that there is a remainder of 14 if you do 14q/21 and a remainder of 4 if you do 4/21. So that tells you it doesn't divide and you don't need to probe deeper? Same logic for the second statement..

I'm not sure everyone would agree that the solution is as obvious as you suggest, especially since only 55% of students answered it correctly.
That said, I may be guilty of over-explaining.

Cheers,
Brent

I was just double checking that I wasn't over-simplifying and lucky to get the answer. Thanks!

Reframing the question

21q+0 = X ? #Is X is divisible by 21 (q - quotient , 0 is the remainder) ------------------------------1

Statement 1 says

14q+4 = X -----------------------------------------------------------------------------------------------2

1=2 implies 14q+4 = 21q

solving the above will say that q cannot be a whole number so the answer will always be NO

so we are down to AD

Statement 2 says

15q+5 = X -----------------------------------------------------------------------------------------------3

1=3 implies 15q+5 = 21q

solving the above will say that q cannot be a whole number so the answer will always be NO

So we are down to D now

Is the positive integer X divisible by 21?

When X is divided by 14, the remainder is 4
When X is divided by 15, the remainder is 5

The rule here to Find the X is using LCm methods. Since 14-4 = 15-5 = 10

Hense

x= LCM (14,15)-10
X= 210-10 => 200 Hense X is 200.

When you divide 200 by 21, it will not divide. Hence option C is satisfied.

Please correct me if I am doing something wrong.

Thanks.

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