Asked: If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

175 =km + 7; where m is an integer
168 = km = 2^3×3×7
Since division by k, leaves remainder 7; k>7


(1) The product of any two factors of k is odd
k is odd; k>7 and km=168
k =21 is the only solution since k is odd, k>7 and I is a factor of 168.
SUFFICIENT

(2) The sum of any two factors of k is even.
k is odd since sum of 2 odd numbers is even, k cannot be even since 1 is a factor of k and is odd.
k is odd; k>7 and km=168
k =21 is the only solution since k is odd, k>7 and I is a factor of 168.
SUFFICIENT

IMO D

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175 = qk + 7
=> qk = 168 = 2^3 * 3 * 7 where q = quotient and k > 7

st1) If product of any two factors is k, all factors of k are odd , so from the factorization above we know k = 21 (since k < 7) SUFFICIENT

st2) If the sum of any two factors is even, all factors of k are odd; so can be deduced similar to st1 SUFFICIENT

The answer should be D, IMO.

A

Both statements are sufficient.

175 = kp + 7

That means kp = 168

Factors of 168 = 2*2*2*3*7

1. That means number is odd and no even factor can't be there.

Only 21 is possible.

2. Again factors can be only even or only odd. So that means either 8 or 21. Not sufficient.

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If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

Except 1 and 7, K will be a factor of 175 - 7 = 168, which will keep a remainder of 7 when divides 175. It may be 84, 42, 21, 14. K has to be greater than 7.

(1) The product of any two factors of k is odd...we can infer that k has no even factor. Among the possible choices only 21 fulfills the condition. Sufficient.
(2) The sum of any two factors of k is even. k has to contain all even or odd factors. But since 1 is factor of all my integers, only 21 is the possibility. Sufficient

D is the answer.

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Given - 175= Kq + 7
168 = kq
2³*7*3= kq

(1) The product of any two factors of k is odd

This says that k is odd

k can be 3, 7 or 21

No definitive answer . Hence INSUFFICIENT.

(2) The sum of any two factors of k is even.

This says that k is odd or even( no odd factors).

k can be 3, 7 21, 2,4,8


Insufficient

(1) & (2) - This says that k is odd

Again insufficient. It conveys the same information as statement (1)

Answer E

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175/k = remainder 7
so the number is divided by 168
prime factors : 2,2,2,3,7
number has to be more than 7)

1:
any 2 factors product is odd= so the only option left is 7*3= 21
when 21 is divided by 175, we get remainder 7
no other factors can have multiple as odd

sufficient


2: it can be
14*6 are factors of 168
if add make sum 20 - even
number becomes 84 leaves remainder of 7
k=84
sum of any 2 factors of 84 should be even= 7,12( become oddd)-reject

but if we add: 7+3 then even and 7.3
k=21

( sum of any 2 factors = 7+3) even

sufficient (k=21)


hence D

175 = Nk + 7

This means k * N = 168.

1. Product of any two factors is odd.
This means all factors of k is odd. k should be odd. k can be 3, 7, 21.
Insufficient.

2. The sum of any two factors of k is even.
This means all factors are odd or all factors of k are even. Latter is not possible, since 1 is a factor. So, all factors are odd. Again, k can be 3, 7, 21. Insufficient.

Answer: E.

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?
=> Possible values for k are 182,357,532,707..

(1) The product of any two factors of k is odd
-- Considering statement 1 alone,
=> For product of any 2 factors to be odd, none of the factor should be even. Or in other words number should be odd.
=> possible choices = 357, 707 ..
Not sufficient.

(2) The sum of any two factors of k is even.
-- Considering statement 1 alone,
=> For sum of any 2 factors to be even, none of the factor should be even. Or in other words number should be odd.
=> possible choices = 357, 707 ..
Not sufficient.

Combing both the statement will not help. IMO answer is E.

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

Given
175=mk+7
K>7
Mk=168

(1) The product of any two factors of k is odd
M=1 k=168
M=2 k=84
M=3 k=56
M=4 k=42
M=6 k=28
M=7 k=24
M=8 k=21
M=12 k=14
M=14 k=12
M=21 k=8
These are all possible values.
To have a product of any two factors of k odd, the number k should be odd. There is only one value that satisfy this. K=21. Sufficient
(2) The sum of any two factors of k is even.
It could 21 or 2 in any power as well. For instance, 8.
Not sufficient

The answer is A

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If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?


(1) The product of any two factors of k is odd
(2) The sum of any two factors of k is even.

given 175 = K*x+7
k would be 168 in this case and its factors ; 2^3*3*7

#1
The product of any two factors of k is odd
possible only when 3*7 ; 21
sufficient
#2
The sum of any two factors of k is even.
possible only at (3,7) factor would be 21
sufficient
OPTION D

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If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

we have 175 = k*x + 7
k*x = 168
\(168 = 2^3 * 3 * 7\)
possible factors: 21 and 8 (other factors 1,2,3,4, 7 are <= remainder 7, hence not possible)

(1) The product of any two factors of k is odd
- The leaves only 21 as the possible answer; product of factors of 8 is always even
SUFFICIENT

(2) The sum of any two factors of k is even.
- This also leaves only 21 as possible answer, because the condition is not true for 8 + 1 = 9
SUFFICIENT

Answer D

k is a positive integer, 175/k remainder 7 so k is a factor of 175-7= 168 , and k must >7 (168= 3*7*2*2*2)
1, Product of any 2 factor of k is odd, so k is an odd number => k=21 => sufficient
2, The sum of any two factor of K is even , so K must be odd ( because 1 is a factor of K )=> k=21=> sufficient
Hence D

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

Given = 175 = k*m + 7 where m is a constant, let see what value k can take or ideally
175 - 7 = 168 has how many factor
168 = 2^3*3*7, we have 16 factor for 168 out of which we have only 4 odd factors (1,3,7, and 21) rest are even
and since we are seeing number factor should be greater than 7 to get a remainder of 7
lets see the statements

(1) The product of any two factors of k is odd
Sufficient : the only value for k would 21 and product of any factor of k remains odd as well

(2) The sum of any two factors of k is even.
Insufficient :k could be 8, sum of any 2 factor is even
k could be 21, again sum of any two factor is even

IMO A

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?
From Given Data
We Know
175= KX+7

So From Here we know K should be Greater than 7

also
we know
175-7=KY
So
KY=168

Possible Values

84 X 1
56 X 3
42 X 4
28 X 6
24 X 7
21 X 8

So K can Have all values greater than 7

This means

K can be 8,21,24,28,42,56,84

(1) The product of any two factors of k is odd

If Product of any two Integer is odd K should be Odd integer itself

Hence K=21

Statement 1 alone is sufficient

(2) The sum of any two factors of k is even.

This means all the factors should be odd
It will only be possible when K is an odd integer
as if K is even integer
For Eg:
if K is 24 which has factors - 1,2,3,4,6,8,12,24
Then 1+24 will be odd
But When K is 21 which has factors- 1,3,7,21

Then by selecting any 2 values we will have odd sum


Hence K cannot be even integer and K= 21
Statement 2 is sufficient


Answer: D

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If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?


(1) The product of any two factors of k is odd- Sufficient. since the only value that when divides 175 leaves remainder 7 and whose product of any two factors is odd is 21.
(2) The sum of any two factors of k is even.- Sufficient. The only number that satisfies the given condition and leaves remainder 7, is 21.

hence, D is the answer.

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

\(175=k( )+7\) & \(k>7\)
\(k( )=168\) so we can infer that k is a multiple of 168(factor form \(168=3x7x(2^3)\))

(1) The product of any two factors of k is odd

\(k = 21(3x7)\) as\( k>7 \)and cant have even factors(any two)

SUFFICIENT

(2) The sum of any two factors of k is even.

sum of two factors is even so either both even or odd

only possible value is 21(its factors are off so sum is even)

also 8 can be considered but factors 2+4=even and 1+4 is odd

SUFFICIENT

SO D is answer

If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

(1) The product of any two factors of k is odd
(2) The sum of any two factors of k is even.

So we have \(\frac{175}{k}\) giving us a remainder of 7.

\(175=k*d+7\)
\(k=\frac{168}{d}\)

Now we takes values of d such that we get k as a positive integer as mentioned in the question.

d=1, k=168
d=2, k=84
d=3, k=56
d=4, k=42
d=6, k=28
d=8, k=21
d=12, k=14
d=14, k=12

(1) Says that product of the factors of k is odd, but we see that factors of k=84(42*2) and k=21(7*3) is both even and odd respectively.
Hence insufficient.

(2) Sum of "any" two factors is even: Here we again take the example of k=21 where the sum of factors is odd, but on the other hand k=84 has sum of factors which is even. Hence we cannot determine the value of k -> Insufficient

Hence the answer is E.

This means 175-7 is divisible by k. This implies k is a factor of 168.
Now, 168 = 2^3 x 3 x 7
k can be any combination of these


This means k doesn't have any even factors. Which means k is itself an odd number.
So, k can be 3, 7 or 21(i.e. 3x7)
Hence statement 1 alone isn't sufficient to give us the value of k


this means the factors of k are either all odd or all even.
k can still be what we got from statement 1 (i.e 3, 7 or 21), in addition to also being 2, 4, or 8.
Again, statement 2 alone is not sufficient to find k

From statement 1 and 2 together, we get:
K has to be ODD. But, that still leaves us with 3 choices (3,7, 21).
Hence, the 2 statements together are also insufficient to determine the value of k.

Answer: E

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If k is a positive integer and 175 divided by k leaves a remainder of 7, what is the value of k?

Solution:
If remainder is 7, value of k>7.
Also, k divides 168 i.e 175-7

(1) The product of any two factors of k is odd

Statement (1) suggests that k has odd factors only. This means that k is either a prime or is a number with all odd factors only.
21 is the only number which fits this as 168 = 21*8 and favors of 21 are 1,3,7,21.

Statement (1) gives value of k.

(2) The sum of any two factors of k is even.

Statement (2) suggests that k has odd factors only or even factors only.

Two numbers fit this condition 21 which has all odd factors and 8 which has all even factors.

Statement (2) does not give unique value of k.

Answer should Option A.

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