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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
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(1) k^2 + 1 is a prime number.
(2) 2k is divisible by 5
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Bunuel
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
Posts: 25
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.
(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.
mialanknox
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Schools: Wharton '26 CBS '26 Kellogg '26 Booth '26
Posts: 136
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.
Posted from my mobile device
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.
Posted from my mobile device
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
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
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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
(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
GMAT 1: 670 Q42 V40
Posts: 24
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
Posted from my mobile device
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
Posted from my mobile device
8.toastcrunch
Joined: 12 Mar 2023
Last visit: 08 Jul 2026
Posts: 101
Location: India
Concentration: Finance, General Management
Products:
Posts: 101
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.
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.
Posts: 10
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.
Bunuel
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.
Bunuel
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
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
_________________
(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
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