GMAT Club World Cup 2022 (DAY 9): If k is a positive integer and k has

the question seems to be flawed
If k is a positive integer and k has 25 distinct positive factors, how many distinct positive factors does 35k have ?

if k has 25 (odd) distinct positive factors than it must be a perfect square
since number of factors = 25 = 1*25 or 5*5 therefore the number k must have two prime factors with power 4
that is for ex 2^4 * 3^4 = 1296 and it has 25 distinct positive factors
or 2^4 * 5^4 and so on so lets assume x^4 * y^4

now from st1
(1) Both 5k and 7k have 18 distinct positive factors.
First case : if 5 was not a factor of k initially then clearly the new prime factoisation of 5k and 7k would entail (5/7)^1*x^4*y^4
therefore no of distinct factors of 5k or 7k is (1+1)*(4+1)*(4+1) = 50
the second case is if 5 or 7 was already one of the factor of k
then no of factors = (5+1) * (4+1) = 30 factors
third case if if both 5 and 7 were factors of k
(5+1) * (5+1) = 36 factors
so we see that in no case we get 18 factors

(2) k is a multiple of 363
here 363 has prime factors 3^1 * 11^2
to make it a perfect square we need odd power of 3 or even power of any other number

I rest my case here.
BTW ..I have chosen A as answer

Correct answer : Choice A

Given k has 25 distinct factors .
To find how many factors 35k has ?

35k = 7*5*k
so we need to find if 7 and 5 are included in the 25 distinct factors or not.

to find distinct factors of a number, we prime factorize it and multiply the power of the prime factors + 1
that is if x = a to the power m * b to the power n
number of factors = (m+1) * (n+1)

statement 1 : says both 5k and 7k have 50 factors
this can happen only if 5 is not a factor k and 7 is not the factor k
=> that is if k = a to the power m * b to the power n
then factors of k = (m+1) * (n+1)
5k = a to the power m * b to the power n * 5 to the power of 1
factors = (m+1) * (n+1) * (1 + 1)
= 25 * 2
= 50
There fore statement 1 clearly says 5 and 7 are not part of k. Hence satement 1 is sufficient

Statemet 2 says k is a multiple of 363
7* 363 is multiple and so is 1 * 363 and so is 5*363
This not good enough to know the number of factors
Hence statement 2 is not sufficient

Correct answer Choice A

I don't know if this is right or not, but worth a shot right?

With the information given in the stimulus, we can deduce that k is a square number since it has an odd number of factors. Also, since the total number of factors is 25, it's the product of either two prime numbers or just one.

So, if we talk about 35k, it can look like one of the following:
a) x^24 * 5 * 7
b) x^4 * y ^ 4 * 5 * 7
c) x^4 * 5^5 * 7
d) x^4 * 5 * 7^5
e) 5^5 * 7^5

The total number of factors in the aforementioned cases is 100, 100, 60, 60 and 36 respectively.

Statement 1:
Both 5k and 7k have 18 distinct positive factors.

Let's take 5k for example.

a) x^24 * 5
b) x^4 * y ^ 4 * 5
c) x^4 * 5^5
d) x^4 * 5 * 7^4
e) 5^5 * 7^4

The total number of factors is 50, 50, 30, 50, and 30.

Hence, this statement is false, imo as it's not possible for 5k and 7k to have 18 distinct positive factors.

Using statement 2:
We can conclude that we are referring to case (b) as 363 = 3 * 11^2. So the value of x and y will be 3 and 11.

Hence, the number of factors 35k will have can be calculated conclusively.

Hence, option B is the answer.

Asked: If k is a positive integer and k has 25 distinct positive factors, how many distinct positive factors does 35k have ?

k is of the form \(p_1^24\) or \(p_1^4p_2^4\)

(1) Both 5k and 7k have 50 distinct positive factors.
Since 5k & 7k both have 50 distinct positive factors, then 5 or 7 are not\( p_1 \)or \(p_2\)
\(35k = 5*7*p_1^24\) ; Number of distinct factors of 35k = 2*2*25 = 100
\(35k = 5*7*p_1^4p_2^4\); Number of distinct factors of 35k = 2*2*5*5 = 100
SUFFICIENT

(2) k is a multiple of 363
\(363 = 2*11^2\)
Since k is a multiple of 363 and k is of the form \(p_1^4p_2^4\); \(k = 3^4*11^4\)
\(35k = 2^4*5*7*11^4\): Number of distinct factors of 35k = 5*2*2*5 = 100
SUFFICIENT

IMO D

We know that k has 25 positive factors.

Hence k can have one of the two forms (p and q are prime numbers) -

Form 1

k = \(p^ {24}\)

Form 2

k = \(p ^ 4 * q ^ 4\)

Asked:

Factors of 5 * 7 * k

Statement 1

Both 5k and 7k have 50 distinct positive factors

Case 1:

k = \(p^ {24}\)

if \(p\neq{5}\)

Number of factors = 2 * 25 = 50

Same analysis is also applicable for 7.

Therefore total factors of 35k = 7 * 5 * \(p^{24}\) = 2 * 2 * 25 = 100

Case 2:

k = \(p ^ 4 * q ^ 4\)

if \(p\neq{5}\) and \(q\neq{5}\)

Number of factors = 2 * 5 * 5 = 50

Same analysis is also applicable for 7.

Therefore total factors of 35k = 7 * 5 * \(p ^ 4 * q ^ 4\) = 2 * 2 * 5 * 5 = 100

Hence A is sufficient

Statement 2

363 = 121 * 3 = \(11^2\) * 3

k = \(11^2\) * 3 * some factor

We know that k has 25 factors, so form 1 is not possible here. Which leaves us with form 2.

Therefore k is of Form 2

k = \(11^4 * 3^4\)

Therefore number of factors of \(7 * 5 * 11^4 * 3^4\) = 2 * 2 * 5 * 5 = 100

Hence B is also sufficient

IMO D

k has 25 positive distinct factors. So \(k=x^4*y^4\)

Statement 1:
5k has 50 distinct positive factors.
So, \(5k=5*x^4*y^4\)

Similarly, 7k has 50 distinct positive factors.
So, \(7k=7*x^4*y^4\)

This implies that x and y are neither 5 nor 7.

\(35k=5*7*x^4*y^4\)

No. of distinct positive factors of 35k=2*2*5*5=100

Statement 1 alone is sufficient.

Statement 2: k is multiple of 363
\(363=3*11^2\)
We know that \(k=x^4*y^4\)
So, k=3^4*11^4

\(35k=5*7*3^4*11^4\)

No. of distinct positive factors of 35k=2*2*5*5=100

Statement 2 alone is sufficient.

Answer is D.

Answer D for me

Here is a picture of the explanation



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If k is a positive integer and k has 25 distinct positive factors, how many distinct positive factors does 35k have ?

(1) Both 5k and 7k have 50 distinct positive factors.
(2) k is a multiple of 363

Statement 1 - we dont know how many factors in 5k or in 7k

not sufficient

statement 2 - 25 distinct positive factors only possible = 24+1 or (4+1)*(4+1)

so K has = 11 and 3; k = 363*a= 11^2*3*a
a should be 11^2*3^3
so K = 11^4*3^4
now we can find factors of 35K

sufficient

Answer B

k is a positive integer and k has 25 distinct positive factors,
let's consider the unique prime decomposition of k=a1^p1. a2^p2. ... an^pn, where a1, a2, ...an are prime numbers
k has 25 factors then : (p1+1).(p2+1)...(pn+1) =5^2
from this, we can say that k is of the form p^24 or of the form p1^4 . p2^4

(1) Both 5k and 7k have 50 distinct positive factors.
50=2.5^2
k is of the form p^24 or of the form p1^4 . p2^4, so from this, 5k=5.p1^24 or 5k=5.p1^4 .p2^4
if 5k=5.p1^24
from this, p could not be equal to 5 because if it is the case, 5k=p1^5, then number of its factors is 26, whereas according to to statement 1 it is 50, So p1 is different from 5
Same reasoning of 7k
then 35k=5.7.p1^24, then number of factors of 35k is 2 . 2 . 25=100
if 5k=5.p1^4 .p2^4
from this, p1 nor p2 could not be equal to 5 because if it is the case, 5k=5.p1^5.p2^4, then the number of its factors is 60, whereas according to to statement 1 it is 50, So p1 is different from 5
Same reasoning of 7k
so 35k=5.7.p1^4 .p2^4 then number of factors of 35k is 2 . 2 . 25=100
1st statement is sufficient
(2) k is a multiple of 363

then k=N x 363=N x 3 . 11^2
k is of the form p^24 or of the form p1^4 . p2^4
then, N=p^24/(3 . 11^2), since N is an integer and p is prime, then this case is not possible,
So k is of the form : p1^4 . p2^4
N=p1^4 . p2^4/ (3 . 11^2) for N to be positive, then p1 must equal 3 and p2 must equal 11, then we'll find the value of k then the value of 35, then the number of factors of 35k,
sufficent,

Our answer, then, is D

[quote="Bunuel"]If k is a positive integer and k has 25 distinct positive factors, how many distinct positive factors does 35k have ?

(1) Both 5k and 7k have 50 distinct positive factors.
(2) k is a multiple of 363


Given k has 25 distinct positive factors. Now as per the formula if k=m^a*n^b*o^c where m,n,o are prime numbers
Then total Number of distinct factors of k =(a+1)*(b+1)*(c+1)
25 can be represented as 1*25 or 5*5
Thus either a=0,b=24 or a=4,b=4
1)As per statement 1 both 5k and 7k have 50 distinct positive factors.
50 can be represent as 2*25 or 5*10 or 50*1
Now comparing the values against k we can conclude that a=0 ,b=24 (as both 5 and 7 are not present originally in k)
Thus for 35k,number of factors would be 5*7*a^0*b^24
which is (1+1)(1+1)(24+1)=100
Thus 1 is sufficient
(2) k is a multiple of 363
363=3*11*11
As K is a multiple of 363 then k=3^a*11^b
Now as k has 25 distinct factors and a>=1 and b>=2
Thus we can conclude that a=4 and b=4
Therefore total number of factors of 35k=5*7*3^4*11^4
=(1+1)*(1+1)*(4+1)*(4+1)=100
Thus statement 2 is sufficient.
Hence D is the answer.

Number of positive factors of a positive integer P=a^m*b^n*c^o.. is given by (m+1)(n+1)(o+1)..

Here k is a positive integer and k has 25 distinct positive factors. We know factors of 25 are 1,5 and 25. Therefore k must be of the form
k=(a^4)*(b^4) OR k=a^24

Statement 1:
"Both 5k and 7k have 50 distinct positive factors."

5k and 7k both has 50 factors.
Let us consider 5k first and 7k case will be same as they both have same distinct factors.

If k is of the form a^24 then 5k=5a^24
This will have (1+1)(24+1)=50 factors

Also, if k is of the form of (a^4)*(b^4) then 5k=5(a^4)*(b^4)
This will have (1+1)(4+1)(4+1)=50 factors

Now let one of the factors of k be equal to 5, then in that case
k=5^25 => number of factors=25+1=26 (Not possible as given 5k has 50 factors)
and k=5^5*b^4 => number of factors=(5+1)*(4+1)=30 (Not possible as given 5k has 50 factors)

We can prove the same thing for 7k as well.

Thus from this statement we can say 5 and 7 are prime to other factors present in k.

Therefore 35k = 5*7*k can take two forms:
35k=5*7*a^24 => Number of factors=(1+1)(1+1)(24+1)=100
OR
35k=5*7*a^4*b^4 => Number of factors=(1+1)(1+1)(4+1)(4+1)=100
In either case number of factors of 35k comes to 100
So statement is SUFFICIENT


Statement 2
"k is a multiple of 363"

Thus k can be written as k=363*n (n is an integer)
OR k=3*11^2*n
From given we know k must be of the form
k=(a^4)*(b^4) OR k=a^24
Clearly, in this case since we already have 3 and 11 as prime factors, the only possibility k can be is of form a^4*b^4
Thus, k=3^4*11^4.

Therefore 35k=5*7*3^4*11^4
Number of factors of this = (1+1)(1+1)(4+1)(4+1)=100
Hence statement is SUFFICIENT.

K is a positive Number with 25 Distinct Factors= 5*5
K=P^4 * Q^4 (Where P & Q are Prime Number)
If we get to know that PQ is a multiple of 5 & 7 we can easily find the Positive Factors of 35K

Option 1: Both 5k and 7k have 50 distinct positive factors
We can find that 5K =50 & 7K=50 Factors (Provided K has 25 Factors)
This means 5 & 7 are not the factors of K
Hence the Factors of 35k =5*7*K = 2* 2*25 =100 No of positive Factors
Sufficient

Option 2 : K is a multiple of 363
363 = 11^2 * 3 the Factors of K are P^4 & Q^4 hence P is 11 & Q is 3
Hence K doesnt have a factor of 5 & 7 hence we can get the factor of 35K as 5*7*K = 100
[i]Sufficient

Option D

Imo D

Given: k is a positive integer and k has 25 distinct positive factors
Implies that K can be of the form of (a^24) or (a^4 * b^4), in which a & b are prime numbers.
To find: how many distinct positive factors does 35k have ?
Statement 1: Both 5k and 7k have 50 distinct positive factors.
5K: 5* (a^24) or 5 * (a^4 * b^4)
For 5k to have 50 distinct factors, a & b cannot be equal to 5 ----(1)
Similarly for 7k to have 50 distinct factors, a & b cannot be equal to 7 -----(2)
35k = 7* 5* (a^24) or 7* 5* (a^4 * b^4)
Total Factors of 35k = 2*2*5*5 [Since a, b are not equal to 5 or 7 from (1) & (2)]
Sufficient

Statement 2: k is a multiple of 363
Implies K = 11^2 * 3 * y
It is given the k has 25 factors and thus we can deduce that y contains 11^2 and 3^3.
35k = 7*5*11^4 * 3^4
Total factors = 2*2*5*5
Sufficient

IMO D

In A, 5k & 7k lead to 50 factors implies that 5 & 7 are not factors of k, and 5^1 and 7^1 makes a total of (1+1)*(24+1)=50 factors
In B, 25 factors can only be derived from (4+1) * (4+1), so k's only prime factors are 3 and 11.

OA = D

As per the question, k has 25 distinct positive factors

which means \(k = a^24\) or \(k = a^4 * b^4\) , where a,b = any prime numbers

now, to find the prime factorization of \(35k = 5*7* a^{24}\) or \(35k = 5*7*a^4*b^4\)

now if we can find the value of a & b or prove that a & b are neither 5 nor 7 then we can find the answer

statement 1 )

Both 5k and 7k have 50 distinct positive factors.

now if \(k = a^{24}\) ,
then prime factorization of \(5k = 5*a^{24}\) , and \(7k = 7*a^{24}\)

if a=5 or 7 then the prime factorization will be \(5k = 5^{25}\) and \(7k = 7^{25}\) which leads to a total number of factor= 26

but as per the statement, the total number of factors is 50
so this is only possible if the value of k is other than 5 or 7

the same will be the case for \(k = a^4 * b^4\)

based on this statement it can conclude that the value of a & b is neither 5 nor 7
so the prime factorization of \(35k = 5*7*a^{24}\) or \(35k = 5*7*a^4*b^4\)

total number of factors = (1+1)(1+1)(24+1) or (1+1)(1+1)(4+1)(4+1) = 100

that's why statement 1 is sufficient

statement 2)
k is a multiple of 363

now 363 = 3 * 121 = \(3 * 11^2 \)

now k is multiple of 363 and \(k = a^{24}\) or \(k = a^4 * b^4\)

this can only be possible if \(k = 3^4 * 11^4\)

so based on that also we \(35k = 5 * 7 * 3^4 * 11^4\) , which means total number of factor = 100

so statement 2 is also sufficient

so , based on that both the statement are alone sufficient to answer the question

Answer choice (D) IMO

Answer is B

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