How many even divisors of 1600 are not multiples of 16?

Question

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

Kinshook · Accepted Answer

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

1600=(2^6)(5^2) 
Total no of divisors of 1600 = (6+1)*(2+1) = 7*3 = 21
Out of which 1, 5 & 25 are odd divisors 
Total no of even divisors 0f 1600 = 21-3 = 18
 1600 = 16 (2^2)(5^2) 
No of divisors which are multiple of 16 = (2+1)*(2+1) = 3*3=9 all are even 
No of even divisors which are not multiple of 16 = 18 - 9 = 9

Alternatively,
 2^0, 2^4, 2^5 & 2^6  are not allowed
Only  2^1, 2^2 & 2^3  are allowed = 3 ways
 5^0, 5^1 , 5^2 are allowed = 3 ways
Total no of even divisors not multiple of 16 = 3*3 =9

IMO C

shridhar786 · Answer

1600 can be written as

1600 =  2^6 * 5^2 
so total number of factors = (6+1)(2+1) = 7*3=21
total number of odd factors = (2+1) = 3 (in counting odd factors we will live all powers of 2)

total numbers of even factors = 21-3 = 18

1600 =  2^4 ( 2^2 * 5^2 )=16( 2^2 * 5^2 )
total number of factors divisible by 16 =(2+1)(2+1)=3*3=9

total number of even factors not divisible by 16 = 18-9 = 9
C is the answer

ruchik · Answer

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

1600 can be written as 2^6 * 5^2
now even divisors of 1600 which are not multiple of 16 include: 2,2*5,2*25,4,4*5, 4*25, 8, 8*5, 8*25

hence 9
Answer = C

Archit3110 · Answer

FACTORS OF 1600 ; 2^6*5^2 
AND FACTORS OF 16; 2^4 
SO 1600/16 ; 2^6*5^2/2^4 ; 2^2*5^2 ; 3*3 ; 9
IMO C

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

JonShukhrat · Answer

How many even divisors of 1600 are not multiples of 16?

factors of 1600 = 4*4*100 and 16=4*4
Multiples of 16 = 16, 32 , 48.............

When divisors of 1600 is a multiple of 10 or 100 then it may not be a multiple of 16

1600(16*100)-multiple of 16
800(16*5) - multiple of 16
400(16*25) - multiple of 16
200-NOT multiple of 16
100-NOT multiple of 16
50-NOT multiple of 16
40-NOT multiple of 16
20-NOT multiple of 16
10-NOT multiple of 16

When divisors of 1600 is less than 16 it cannot be a multiple of 16
8 - NOT multiple of 16
4 - NOT multiple of 16
2 - NOT multiple of 16

Answer C = 9

FIRST APPROACH : We need to prime factorize 1600: 1600 = 16*100 = 2^6*5^2 Overall number of factors 1600 have is the product of powers increased by 1: (6+1)(2+1)=21 The number of Odd factors is found by removing 2^6 becuase it will make factors even. So we will have 5^2 which has (2+1) = 3 odd factors. The number of Even factors = All factors - Odd factors = 21 - 3 = 18 . So 1600 have 18 even and 3 odd factors. Next, how many of 18 even factors are NOT divisible by 16 or 2^4 ? For a number NOT to be divisible by 2^4 , it must have at most 2^3 as a prime factor. Hence, 2^3*5^2 will give us the number of factors not divisible by 2^4 . So All factors not devisible = (3+1)*(2+1)=12 . We need to remove 3 odd factors: 12-3=9 . Thus 9 even factors of 1600 are not divisible by 16. SECOND APPROACH : For this manual method we again need to prime factorize 1600: 1600 = 16*100 = 2^6*5^2 So even factors not devisibe by 16 would be the product of 2's and 5's we have when the highest possible power of 2 is 3. Let's manually find those factors: 2 2^2 2^3 2*5 2^2*5 2^3*5 2*5^2 2^2*5^2 2^3*5^2 Overall 9. Hence C

Abhi077 · Answer

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

Solution:

We need to focus on EVEN divisors of 1600 only, therefore each divisor there of must be divisible by 2.

Prime factorization of 1600 is  2^6  X  5^2

Prime factorization of 16 is  2^4

We can notice that the powers of 2 less than exponent 4 combining with the powers of 5 less than 3 are satisfying this condition,

Therefore we get a total of 9 numbers  2^1  ,  2^2  ,  2^3  ,  2  X  5 ,  2^2  X  5  ,  2^3  X  5 ,  2  X 5^2  , 2^2  X  5^2 ,  2^3  X  5^2  i.e  2,4,8,10,20,40,50, 100 & 200  that are the even factors of 1600 which CANNOT be the multiples of 16 since the cannot be divided by 2^4 ,

Hence the answer is C

nick1816 · Answer

1600= 2^6*5^2 
Any divisor of 1600 can be written in the form of  2^a*5^b , where 0≤a≤6 and 0≤b≤2

As divisor must be even, exponent of 2 can't be 0. Also, divisor is not multiple of 16, hence exponent of 2 can't be 4,5 or 6.

Number of value a can take=3 {1, 2 or 3}
Number of value b can take=3 {0, 1 or 2}

Number of even divisors of 1600 are not multiples of 16= 3*3=9

AvidDreamer09 · Answer

How many even divisors of 1600 are not multiples of 16?

factors of 1600 = 4*4*100 and 16=4*4
Multiples of 16 = 16, 32 , 48.............

When divisors of 1600 is a multiple of 10 or 100 then it may not be a multiple of 16

1600(16*100)-multiple of 16
800(16*5) - multiple of 16
400(16*25) - multiple of 16
200-NOT multiple of 16
100-NOT multiple of 16
50-NOT multiple of 16
40-NOT multiple of 16
20-NOT multiple of 16
10-NOT multiple of 16

When divisors of 1600 is less than 16 it cannot be a multiple of 16
8 - NOT multiple of 16
4 - NOT multiple of 16
2 - NOT multiple of 16

Answer C = 9

Deepesh2207 · Answer

Prime factorization off 1600 = 2^6 * 5 ^2
Since we have to find out even factors of 1600 not divisible by 16 a.k.a 2^4, we can group such factors as below:
First taking only one 5:
2 * 5 = 10 
2^2 * 5 = 20 
2^3 * 5 = 40

Then pairing 2s with both the 5s:
2 * 5^2 = 50
2^2 * 5^2 = 100
2^3 * 5^3 = 200
And it's clear that in addition to the above, 2, 4 and 8 are the other 3 numbers not divisible by 16.
We can't take any powers of 2 >= 4 as it would be a multiple of 16.
Hence the total number of factors satisfying the problem statement adds up to 9.
Ans:  C

MahmoudFawzy · Answer

1600 = 2^65^2

The number of distinctive divisors =  (6+1)(2+1) = 21 
By taking 5^2 only into account, the number of odd divisors =  (2+1) = 3 
Then the number of even divisors =  21 - 3 = 18

The number of factors divisible by 16 (and they are even for sure) =  (3)(2+1) = 9  (here we calculate the factors of  2^25^2 as we exclude  2^4  as a common factor for them)
Then the number of even divisors of 1600 that are not multiple of 16 =  18 - 9 = 9

C

manass · Answer

1600= 2^6 * 5^2
with each power of 5
exclude 2^0 as we want even numbers.
5^0 ---> 2^1 , 2^2 , 2^3
5^1 ---> 2^1 , 2^2 , 2^3
5^2 ---> 2^1 , 2^2 , 2^3

total 9.

AliciaSierra · Answer

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

IMO answer should be 9 i.e. C.

Factorize 1600

1600 = 2^6*5^2

Total number of divisors of 1600=  (6+1)*(2+1)= 21 
Total number of odd factors of 1600 =  (1)(2+1)= 3 
So even number of factors =  21-3=18

following are even factors.

2 10 50
4 20 100
8 40 200 
16 80 400
32 160 800
64 320 1600

Anything less than 16 is not multiple of 16 so 2,4,8 are not multiples of 16 so corresponding 5 & 25 multiples of 2,4,8 i.e. 10 , 20, 40, 50,100,200 are also not multiple of 16.

Total 9 even divisors of 1600 are not multiples of 16.

GKomoku · Answer

How many even divisors of 1600 are not multiples of 16?

1600 | 2
800 | 2
400 | 2
200 | 2
100 | 2
50 | 2
25 | 5
5 | 5
1 |

2^6 * 5^2 = 1600

So --->

2^1 = 2 - even, not multiple of 16
2^2 = 4 - even, not multiple of 16
2^3 = 8 - even, not multiple of 16
2^4 = 16 / 16
2^5 = 32 / 16
2^6 = 64 / 16
2^1 * 5^1 = 10 - even, not multiple of 16
2^1 * 5^2 = 50 - even, not multiple of 16
2^2 * 5^1 = 20 - even, not multiple of 16
2^2 * 5^2 = 100 - even, not multiple of 16
2^3 * 5^1 = 40 - even, not multiple of 16
2^3 * 5^2 = 200 - even, not multiple of 16
5^1 = 5 - odd
5^2 = 25 - odd

So answer is 9

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

C is the answer

Nups1324 · Answer

Hi Kinshook , In this line of your explanation "No of divisors which are multiple of 16 = (2+1)*(2+1) = 3*3=9 all are even", how all are even.? 1600 =16×2^2×5^2 right.. so it can be 2^0×5^1 which is 5. Which is odd.? Secondly, here you're trying to find out the multiples of 16 right? " 1600 = 16 (2^2)(5^2) No of divisors which are multiple of 16 = (2+1)*(2+1) = 3*3=9 all are even" But then 2^2×5^1=20 is not a multiple of 16. How did you do it.? I'd love to learn how you did it. Thank you

Posted from my mobile device

TarunKumar1234 · Answer

How many even divisors of 1600 are not multiples of 16?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 18

1600 can be written as 2^6 * 5^2
now even divisors of 1600 which are not multiple of 16 include: 2,2*5,2*25,4,4*5, 4*25, 8, 8*5, 8*25

hence 9
Answer = C

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How many even divisors of 1600 are not multiples of 16?

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How many even divisors of 1600 are not multiples of 16?

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