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How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:00
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How many even divisors of 1600 are not multiples of 16? (A) 4 (B) 6 (C) 9 (D) 12 (E) 18
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How many even divisors of 1600 are not multiples of 16?
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Updated on: 15 Jul 2019, 09:24
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 = 213 = 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
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Originally posted by Kinshook on 15 Jul 2019, 08:34.
Last edited by Kinshook on 15 Jul 2019, 09:24, edited 5 times in total.




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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:17
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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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:22
How many even divisors of 1600 are not multiples of 16?
(A) 4 (B) 6 (C) 9 (D) 12 (E) 18
Fairly simple question
You need to look for multiples below 16 first like 2, 4, 8, 10 and then some like 20, 40, 100, 400, etc.
The answer is C 9.



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:32
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



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:34
Answer is B. Even divisor of 1600 are: 1600/2=800, 1600/4=400, 1600/8=200, 1600/16.. 20, 32, 40, 50, 64,80,and 100 then divide those results by 16 and the total Multiples of 16 are 6. Total divisors of 1600 are 12. so 126=6



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How many even divisors of 1600 are not multiples of 16?
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Updated on: 15 Jul 2019, 08:38
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 = 213 = 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 = 189 = 9 C is the answer
Originally posted by shridhar786 on 15 Jul 2019, 08:35.
Last edited by shridhar786 on 15 Jul 2019, 08:38, edited 1 time in total.



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How many even divisors of 1600 are not multiples of 16?
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Updated on: 15 Jul 2019, 09:13
\(1600 = 2^6*5^2\)
2 cannot take the power 0,4,5,6 since then we either won't get an even factor or we will get a factor divisible by 16. So 2's power can 1,2 or 3  3 options 5's power can be 0,1 or 2  3 options
Therefore the total number of even factors which are not divisible by 16 = 3*3 = 9
Answer is (C)
Originally posted by firas92 on 15 Jul 2019, 08:36.
Last edited by firas92 on 15 Jul 2019, 09:13, edited 1 time in total.



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:38
C) 9
16, 32, 64, 80, 160, 320, 400, 800, 1600



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:39
IMO answer is D:
factors of 1600 are: 1 2 4 5 8 10 16 20 25 32 40 50 64 80 100 160 200 320 400 800 1600 out of these removing 16,32,64,80,160,320,800 and 1600, we are left with 12 numbers so D



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How many even divisors of 1600 are not multiples of 16?
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Updated on: 15 Jul 2019, 23:24
1600 = 16*10*10 = 2^3*2*(2*5)*(2*5) Even numbers not multiples of 16 can have a maximum of 2^3 as a factor. 2, 2^2, 2^3, 2*5, 2^2*5, 2^3*5, 2*5^2, (2^2)*(5^2), (2^3)*(5^2) 2,4,8,10,20,40,50,100,200 9 numbers Option C Posted from my mobile device
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Originally posted by prashanths on 15 Jul 2019, 08:39.
Last edited by prashanths on 15 Jul 2019, 23:24, edited 1 time in total.



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:40
A it is , will provide explanation a bit later



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:43
D
Factorize 1600, we get 16000 = 2^6*5^2. So total no of factors can be (6+1)*(2+1) = 21.
Now, estimate the factors that are multiple of 16. Now, 1600 can be written as 1600 = 16*(2^2*5^2)
From above, total possible divisors of 1600, which are also multiple of 16 are (2+1)*(2+1) = 9
So, total factors that are NOT multiple of 16 should be 21  9 = 12.



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How many even divisors of 1600 are not multiples of 16?
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Updated on: 15 Jul 2019, 09:50
How many even divisors of 1600 are not multiples of 16?
1600= 2^6*5^2 Total factors = (6+1)*(2+1)=7*3=21 Odd factors are: 1, 5, 25 Then even factors: 213=18
To find the Number of factors that multiples of 16 we need to exclude it from 1600. Thus we have 100=2^2*5^2 Number of factors that multiples of 16 = (2+1)*(2+1)=9
To find the number of even factors that are not multiples of 16, just subtract 9 from total even factors 189=9
IMO C
Originally posted by ancored on 15 Jul 2019, 08:44.
Last edited by ancored on 15 Jul 2019, 09:50, edited 1 time in total.



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:45
1600 = 2^6*5^2 No of Factors of 1600 = (6+1)*(2+1)=7*3=21 factors Now, 3 factors multiples of 5 are definitely not multiples of 16 In case of multiples of 2, 2,4&8 are not multiples of 16 but 2^4 onward are multiples of 16. So 3 factors here as well Now 3*3=9 factors are not multiple of 16 but remaining 4*3=12 are multiples of 16. So 9 is the answer, hence C
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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:46
We can find the answer by using this
Number of divisors =(2^1+2^2+2^3)(5^0+5^1+5^3) =(3)*(3) =9
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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:47
How many even divisors of 1600 are not multiples of 16? 1600 is not divisible by 3 and any multiple of 3 Even factors of 16 lower than 16 must be multiples of 1600 so 2,4,8 (3 factors other than 1 and 16 itself) 1600 can be factorized into 2^3*5^2*2^3. So factors of 200 that are factors of 1600 are 10,20,40,50,100,200 (5*2,25*2,5*4,5*24,5*8,25*8 based on prime factorization of 200) IMO C  9  2,4,8,10,20,40,50,100,200.
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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:47
How many even divisors of 1600 are not multiples of 16? (A) 4 (B) 6 (C) 9 (D) 12 (E) 18 Factors of 1600 are  \(2^6 * 5^2\) \(16 = 2^4\) There fore maximum powers of 2 can be only 3. Total number of factors such that 1600 are not multiples of 16 = Number of factors of \(2^3 * 5^2\) = (3+1) * (2+1) = 12. Total number of even factors such that 1600 are not multiples of 16 = Number of even factors of \(2^3 * 5^2\) = (2+1) * (2+1) = 9. Ans  C
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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:50
1600 = 2^6 X 5^2 not divisible by 16 implies the max power of 2 will be 3.
==> 2^3 X 5^2 number of total factors for this is (3+1)(2+1) = 12 12 factors include (1,5,25) as factors too.
Remove 3 odd factors 12 3 = 9 even factors that are not multiple of 16.
Hence, C is the answer.



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Re: How many even divisors of 1600 are not multiples of 16?
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15 Jul 2019, 08:55
Prime factorization 1600 is 2^6 x 5^2 No of factors not divisible by 16 which has prime factorization as 2^3 x 5^2 is 4x3 = 12
No of odd factors in the above factorization are 3
No of even factors which are not multiples of 16 are 123 = 9
C is correct.




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