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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?  [#permalink]

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15 Jul 2019, 08:00
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53% (01:40) correct 47% (01:51) wrong based on 243 sessions

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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

 This question was provided by Veritas Prep for the Game of Timers Competition

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How many even divisors of 1600 are not multiples of 16?  [#permalink]

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Updated on: 15 Jul 2019, 09:24
3
2
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
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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?  [#permalink]

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15 Jul 2019, 08:17
1
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
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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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.

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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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15 Jul 2019, 08:32
2
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?  [#permalink]

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15 Jul 2019, 08:34
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 12-6=6
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How many even divisors of 1600 are not multiples of 16?  [#permalink]

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Updated on: 15 Jul 2019, 08:38
3
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

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?  [#permalink]

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Updated on: 15 Jul 2019, 09:13
1
$$1600 = 2^6*5^2$$

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

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?  [#permalink]

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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?  [#permalink]

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15 Jul 2019, 08:39

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?  [#permalink]

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Updated on: 15 Jul 2019, 23:24
1
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

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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?  [#permalink]

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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?  [#permalink]

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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?  [#permalink]

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Updated on: 15 Jul 2019, 09:50
1
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: 21-3=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
18-9=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?  [#permalink]

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15 Jul 2019, 08:45
1
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?  [#permalink]

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15 Jul 2019, 08:46
1
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?  [#permalink]

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15 Jul 2019, 08:47
1
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?  [#permalink]

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15 Jul 2019, 08:47
1
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?  [#permalink]

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15 Jul 2019, 08:50
1
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.

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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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15 Jul 2019, 08:55
1
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 12-3 = 9

C is correct.
Re: How many even divisors of 1600 are not multiples of 16?   [#permalink] 15 Jul 2019, 08:55

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