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

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.

Yes K is a multiple of 10

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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

1. k^2+1 is a prime number
k can be 1,2 -> 2,5 (not multiple of 10)
k can also be 10 -> 101 (multiple of 10)
Insufficient

2. 2k is divisible by 5
k = 5 -> 10 (not a multiple of 10)
k=10 -> 50 (multiple of 10)
Insufficient

Both together are sufficient as only possible values would be 10,20,30..etc

Therefore answer is C

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K is a multiple of 10

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

B statetment 2 alone is sufficient

C, both are required

A is correct,
substitute k by 1,2,4,6,9,14 --> not a multiple of 10
B substitute k by 5 Hence 5 is not a multiple of 10 but substitute K by 10 it is the multiple of 10.

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

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

true for multiple values eg 1,2,3,10
Not sufficient


(2) 2k is divisible by 5

therefore k must be multiple of 5.

Correct answer Option B

I vote for B.
A is insufficient as many numbers fulfill the criteria in the stem.
B is sufficient

2 alone is sufficient

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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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Lets start with s2; 2k is divisible by 5. for 2k to divisbile by 5, k needs to be a multiple of 5. Hence value of k will start from 5, 10, 15, 20 and so on... so k may or may not be multiple of 10. NS

S1; k^2 + 1 is a prime number. Since k is positive number, lets start with the minimum value of k possible = 1.
1^2 + 1 = 2
2^2 + 1 = 5
3^2 + 1 = 10 (cant be included since its not prime.)
4^2 + 1 = 17
6^2 + 1 = 37
10^2 + 1 = 101


from S1 k has multiple values, in which one of them is multiple of 10, hence NS
combing both only one common value satisfies both i.e k = 10. Hence sufficient.

NOTE: from S2 we know that k is a multiple of 5 and every multiple of 5 square will either end in 5 or 0
from s1 we need k^2 +1 to be prime. and that is only possible when k^2 (which is a multiple of 5 square) will end in 0. Hence all those value must be multiple of 10 as well.

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.

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

(1) k^2 + 1 is a prime number.
if k^2+1 is prime then k has to be even
Not sufficient on its own

(2) 2k is divisible by 5
if 2k/5 then k is divisible by 5
Not sufficient on its own

Hence for k to be a multiple of 10, both have to be valid.
Hence C

1. We can take k=6 or k=10. Both satisfy the first condition. Hence A is not sufficient.
2. We can take k=5 or k=10. Both satisfy the first condition. Hence B is not sufficient.
Taking both together, only those values of k will satisfy the equation that are multiple of 10. Multiple of 5 but not 10 (from the second scenario) will always give even number (grater than 2, as k is positive integer), hence it will not be prime.
The only way to make both sentences work is to take k as multiple of 10. Hence C is answer.