Around the World in 80 Questions (Day 1): If k is a positive integer

Question

If k is a positive integer, is k a multiple of 10 ?

(1) k^2 + 1 is a prime number.
(2) 2k is divisible by 5

mialanknox · Accepted Answer

Q: Is K a multiple of 2 and 5?

Statement 1: K^2 +1 = prime 
Then k^2 = prime -1 (prime nos 1,3,5,7,11, etc..)
then k^2 must be even. 
That is k is a multiple of 2. Value of k could be 2,4,6,8,10. We cannot say if k is a multiple of 5. Therefore, Statement 1 insufficient.

Statement 2: 2k/5 = integer
2 is not divisible by 5. Therefore, k must be divisible by 5. 
K is a multiple of 5. Values of 5 be with a unit digit of 0,5. But we cant say all values of 5 will be multiple of 2. Therefore, Statement 2 insufficient.

Combined, both statements are sufficient. Therefore, option C is the correct answer.

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missionmba2025 · Answer

Statement 1 :

k = 10 ; k^2 + 1 = 101

k is a multiple of 10

k = 2; k ^2 + 1 = 5

k is not a multiple of 10

Not sufficient

Statement 2 :

k = 10

k is a multiple of 10

k = 5;

k is not a multiple of 10

Not sufficient

Combined

k^2 + 1 must be odd.

k^2 is even, hence k is even. Thus k must have a 2 and 5.

k is a multiple of 10.

IMO C

Abhishek_007 · Answer

Let's analyze each statement:

(1) k^2 + 1 is a prime number.
If k is a positive integer, then k^2 is also a positive integer, and adding 1 to it will still be a positive integer. For k^2 + 1 to be a prime number, it means that k^2 + 1 cannot be divisible by any integer other than 1 and itself.

Example:
- When k = 1, k^2 + 1 = 2, which is a prime number.
- When k = 2, k^2 + 1 = 5, which is a prime number.
- When k = 3, k^2 + 1 = 10, which is not a prime number.

Based on the above examples, we can see that statement (1) is not sufficient to determine if k is a multiple of 10.

(2) 2k is divisible by 5.
If 2k is divisible by 5, it means k is divisible by 5. In other words, k is a multiple of 5.

Example:
- When k = 5, 2k = 10, which is divisible by 5.
- When k = 10, 2k = 20, which is divisible by 5.

Based on the above examples, we can see that statement (2) is sufficient to determine if k is a multiple of 10.

Combining both statements:
By combining the two statements, we know that k is divisible by 5 and that k^2 + 1 is a prime number. However, statement (1) doesn't give us enough information to determine if k is a multiple of 10, so combining both statements won't help either.

Therefore, the answer is that we cannot determine if k is a multiple of 10 with the given information.

turbojuly · Answer

1) K^2+1 is prime.

Try k = 2 -> k^2+1 = 5, prime; k is not a multiple of 10
Try k = 10 -> k^2+1 = 101, prime; k is a multiple of 10

Exclude A, D.

2) 2k is divisible by 5, is k a multiple of 10?

k = 5 -> k is not a multiple of 10
k = 10 -> k is a multiple of 10.

Exclude B.

1) and 2) together

2k is divisible by 5 means that k is divisible by 5, k = 5n.
k^2 + 1 is prime - is k a multiple of 10?

n = 1 -> k=5, k^2+1 = 26 not prime
n = 2 -> k=10, k^2+1 = 101 prime -> k is a multiple of 10
n = 3 -> k=15, k^2+1 = 226 not prime
n = 4 -> k=20, k^2+1 = 401 prime -> k is a multiple of 10
n = 5 -> k =25, k^2+1 = 626 not prime
n = 6 -> k = 30, k^2+1 = 901 prime -> k is a multiple of 10

overall, we can see that:
n can be odd or even
if n is odd (1, 3, 5...), we get a number ending in 5 as k^2, we add 1 we get ending in 6 and it is not prime
if n is even (2, 4, 6), we get a number ending in 0 (which is not prime), but we add 1, and that number is not divisible by any prime number between 1 and k.
therefore, n is even, and when is even, k is a multiple of 10. C is the correct answer

rxb266 · Answer

If k is a positive integer, is k a multiple of 10 ?

(1) k^2 + 1 is a prime number.

If K is 20 .... k^2 + 1 = 401, which is a prime no.  YES - k is a multiple of 10 
If K is 2 .... k^2 + 1 = 5, which is a prime no.  NO- k is NOT a multiple of 10

INSUFFICIENT

(2) 2k is divisible by 5

Possible values of K can be 10 ,5,15,20 .... k may or may not be a multiple of 10

INSUFFICIENT

Combining 1 and 2

k^2 + 1 is a prime number and 2k is divisible by 5.. We can infer from both the statements that k must be even  (for k^2 + 1 to be a prime number)  and a multiple of 5  (since 2k is divisible by 5)

Hence: Answer Option C

khannadwija311 · Answer

In this question, we wish to check if K = 10A

Statement 1: k^2+1 is prime

Case 1: if k=2, k^2+1=5; prime and 2 is not a multiple of 10
Case 2: if k=10, k^2+1=101; prime and 10 is a multiple of 10

So, not sufficient

Statement 2: 2k is divisible by 5. Only numbers of the form 2k that are divisible by 5 are 10,20,30 and so on…
So, all these would be ending with 5 I.e 5,10,15 and so on…
Where 10 is a multiple of 10 but 15 is not.

Statement 1+ Statement 2

K=10,20 
Statement 1 and 2 both follow

K=2, 15 only one follows

So, answer is C

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8.toastcrunch · Answer

Statement 1:  If k = 1, then k^2 + 1 = 2, which is prime. However, if k = 2, then k^2 + 1 = 5, which is prime. If k = 3, then k^2 + 1 = 10, which is not prime. Therefore, statement 1 is not sufficient to determine whether k is a multiple of 10.

Statement 2:  If k is divisible by 5, then 2k is also divisible by 5. However, if k is not divisible by 5, then 2k is also not divisible by 5. Therefore, statement 2 is not sufficient to determine whether k is a multiple of 10.

Statements 1 and 2 combined:  If k is odd, then k^2 + 1 is even and therefore not prime. Therefore, k must be even. If k is even, then 2k is divisible by 10 if and only if k is divisible by 5. Therefore, statements 1 and 2 together are sufficient to determine whether k is a multiple of 10.

The answer is C , statements 1 and 2 together are sufficient to answer the question.

ankemmrudula · Answer

Statement 1 alone doesn't answer because k can be a multiple of 10 (for say k^2 +1 =101; k=10 but for k^2+1 = 3, k=sqrt 2). hence insufficient. Statement 2 alone is insufficient as 2k is divisible by 5 , i.e k can be 5 or 10. But together it gives values of k =multiples of 10.

BottomJee · Answer

We need to find whether k = 10*n; where n >=1
 From Statement 1  we get, 
 K^2 + 1  = Prime no. = odd no. 
Since we know that Even + Odd = Odd, then  K^2  has to be Even. This implies that k is even. 
We can take some examples to check our theory.
 2^2 + 1 = 5  = odd no. = prime
 4^2 + 1 = 17  = odd no. = prime
 6^2 + 1 = 37  = odd no. = prime
 10^2 + 1 = 101  = odd no. = prime

So k can be a multiple of 10 or it cannot, but it will always be Even. This knowledge by itself is not sufficient to answer.

From Statement 2  we get, k =  5^n  * a * b * c * ...; where n>=1 & a,b,c could be any integers which can have any amount of power raised to them.

This implies that k = 5/25/125/... or 10/15/200/ 30....and it can go on and on. This statement by itself is not sufficient.

Considering Statements 1 & 2 together, we get
 k has atleast one 5 as its multiple and k is also even. Therefore, minimum value of k has to be equal to 10.
Hence, K is a multiple of 10.

IMO OA could be C.

Nabneet · Answer

Statement 1: 
If k=10, k^2 + 1 =101 Prime no. --------- K multiple of 10 YES
k=1 . k^2 + 1 =2 Prime no. --------------- K multiple of 10 NO

NS

Statement 2: 
K=5, 2k is divisible by 5 ........... K multiple of 10 NO
K=10, 2k is divisible by 5 ........ K multiple of 10 YES

NS

Combine:
K can take values ending with 5 or 0 (because of st 2)
 
K^2 + 1 to be prime, K needs to be even no. 
even +1 = odd

So the values that K can take will end with 0.

So K always multiple of 10. Sufficient

Answer : C

pintukr · Answer

If k is a positive integer, is k a multiple of 10 ?

(1) k^2 + 1 is a prime number. 
for k=2, k^2 + 1 =5 is a prime number. -----> (NO, k is not a multiple of 10)
for k=10, k^2 + 1 =101 is a prime number. -----> (YES, k is a multiple of 10)
 
 clearly , INSUFFICIENT

(2) 2k is divisible by 5 
for k=5, 2k is divisible by 5. -----> (NO, k is not a multiple of 10) 
for k=10, 2k is divisible by 5. -----> (YES, k is not a multiple of 10) 
 
 clearly , INSUFFICIENT

(1) & (2) together 
 k is a multiple of 5 (Using (2) )
also, k^2 + 1 is a prime number (Using (1) )

hence, k will be a even multiple of 5 ----> k will be a multiple of 10
YES, k is a multiple of 10

SUFFICIENT...since, YES is the unique answer

(C) is the CORRECT answer

Around the World in 80 Questions (Day 1): If k is a positive integer

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