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How many even divisors of 1600 are not multiples 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, 19:36
How many even divisors of 1600 are not multiples of 16?
how many factors of 1600 are not multiples of 16 or are not divisible by 16
Factors of 1600 = $$2^6$$ * $$5^2$$ = 21 factors
1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800, 1600

How many numbers are divisible by 16

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800, 1600,

Total numbers not divisible by 16 - 12

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

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15 Jul 2019, 20:02
answer is 4 as 1600 is 2^6 and 5 ^2, hence 2, 4, 8 and 10, 20 and 40 are not multiples 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, 20:29
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 = $$2^6 *5^2$$

now for multiple of 16
min power of 2 should be 4
hence any divisor with power less than 4 of 2 will not be acceptable
Also given it should be even thus there should be at-least 2

so possible such numbers will be of the form
$$(2^1 +2^2 + 2^3)(5^0 +5^1 +5^2)$$

total such divisors will be (3) *(3)
= 9
Thus 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, 20:48
1
1600=2^6 * 5^2

Total possible factors(or divisors)= (6+1)*(2+1) = 21
Odd factors of 1600 are 1, 5 and 25.

Hence total even divisors of 1600= 21-2 = 18

Let's list the multiples of 16 or 2^4
$$\frac{2^6*5^2}{2^4}$$ => 2^2 * 5^2

Possible factors= (2+1)*(2+1) = 9

(16,32,64,80,160,320,400,800,1600)

Hence option C.
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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, 21:41
1
Prime Factorisation of 1600,
1600=$$2^{6}$$*$$5^{2}$$
No. of factors of 1600 = 7*3=21

No. of even factors of 1600(Reduce the power of 2 by 1 and find the no. of factors) = 6*3=18

But 16=$$2^{4}$$
Therefore,
No.of factors of 1600 which are multiples of 16= No. of factors of $$2^{2}$$*$$5{2}$$ = 3*3=9
Thus,
No.of even factors of 1600 which are NOT multiples of 16= 18-9=9

Ans. C
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Originally posted by aarkay on 15 Jul 2019, 21:32.
Last edited by aarkay on 15 Jul 2019, 21:41, 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, 21:36
1
How many even divisors of 1600 are not multiples of 16?

1600 = 2^6 * 5^2

We are looking for even divisors, so 2^0 can not be there.
And it should not be a multiple of 16 so 2^4/5/6 can not be there.

We can take 2^1/2/3 and 5^0/1/2

So the answer will be: 3 * 3 = 9

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

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15 Jul 2019, 21:47
1
One way to approach this problem is find the prime factorization of 1600. I split 1600 into 16x100

16=2^4 and 100= (2^2)*(5^2)
So 1600=(2^4)*{(2^2)*(5^2)}
Even factors 1600 which are not multiples of 16 are all even factors less than 16 and even factors greater than 16 but not divisible by 16.
All factors of 1600 greater than 200 are multiples of 16. Hence the goal is to find even factors of 1600 up to 200 which are not multiples of 16. This yields the ff set:
{2,4,8,10,20,40,50,100,200}
There are therefore 9 even factors of 1600 which are not multiples 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, 21:56
1
How many even divisors of 1600 are not multiples of 16?

Here, divisor can have maximum 3 as power of 2 and minimum 1. (Because it is even.)

1600 = 2^6 * 5^2

total possibilities for the power of 2 = 3 (1, 2, 3)
total possibilities for the power of 5 = 3 (0, 1, 2)

So, total even divisors satisfying the given condition = 9

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

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Updated on: 16 Jul 2019, 00:19
1
How many even divisors of 1600 are not multiples of 16?

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

Soln:

Divisors of 1600 are

1,2,4,5,8,10,16,20,25,32,40,50,64,80,100,160,200,320,400,800,1600

Even divisors of 1600, that are not multiple of 16 are
2,4,8, 10,20,40,50,100,200

Correct Ans C

Originally posted by komals06 on 15 Jul 2019, 23:57.
Last edited by komals06 on 16 Jul 2019, 00:19, edited 4 times in total.
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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16 Jul 2019, 00:03
How many even divisors of 1600 are not multiples of 16?

1600 is divisible by 16 ,
==> 16*10*10
= 16 * 2^2 * 5*2

So total multiple of number with out 16 = (2+1)(2+1) = 3*3 = 9
Odd divisors = (2+1) = 3
even divisors = 9-3 = 6

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

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16 Jul 2019, 01:19
How divisors does 1600 have?

1600 = 2^4 * 2^2 * 5^2 = 2^6 * 5^2 = 7 * 3 = 21

Now how even divisors does 1600 have?

We know 1600 = 2^6 * 5^2
=> 5, 25, 1 are the only odd divisors of 1600
=> 1600 has 18 even divisors

Now, which divisors of 1600 are not multiples of 16?
=> This would include any divisor which does not have a 16 or 2^4 in it's powers.

Lets list these out as the number is small

1 -> We saw above.
5 -> We saw above.
25 -> We saw above.
2 -> Divisor of 1600 and does not have 16 in it.
4 -> Divisor of 1600 and does not have 16 in it.
8 -> Divisor of 1600 and does not have 16 in it.
2 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 5 in it as 5 is also a divisor of 1600.
4 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 5 in it as 5 is also a divisor of 1600.
8 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 5 in it as 5 is also a divisor of 1600.
2 * 5 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 25 in it as 25 is also a divisor of 1600.
4 * 5 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 25 in it as 25 is also a divisor of 1600.
8 * 5 * 5 -> Divisor of 1600 and does not have 16 in it. Adding a 25 in it as 25 is also a divisor of 1600.

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

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16 Jul 2019, 01:28
1
To solve this problem, we need to apply prime factorization here. 1600 can be written as 2 to the power of 6 and 5 to the power of 2. Since we need even factors that are not multiples of 16, we can exclude 2 to the power of 4, 2 to the power of 5, 2 to the power of 6, 5 to the power of one, and 5 to the power of two. Overall factors are (6+1)*(2+1)=21. We will subtract multiples of 16, which are 1, 2^4, 2^5, 2^6, 5^1, 5^2, 2^4*5^1, 2^4*5^2, 2^5*5^1, 2^5*5^2, 2^6*5^1, and 2^6*5^2, totally 12 factors. 21-12=9 (C)
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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16 Jul 2019, 01:43
1
Formula: If $$N = 2^a*3^b*5^c*7^d*$$ . . . . .
Number of total divisors = (a + 1)*(b + 1)*(c + 1)*(d + 1)* . . . .

$$1600 = 2^6*5^2$$
Total divisors of 1600 = (6 + 1)*(2 + 1) = 7*3 = 21
Number of Odd divisors of 1600 = 1, 5, 5^2 = 3
--> Number of even divisors of 1600 = 21 - 3 = 18

Even divisors that are multiples of 16 = Number of total divisors of 100 [1600 = 16*100]
--> Total divisors of 100 = $$2^2*5^2$$ = (2 + 1)*(2 + 1) = 3*3 = 9

So, Even divisors of 1600 that are not multiples of 16 = 18 - 9 = 9

IMO Option C

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

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16 Jul 2019, 02:51
A good question in my opinion.

We can write 1600 in prime factorization form as 2^6 x 5^2.
We need to find number of factors which are even but not a multiple of 16 I. E. The power of 2 Shoulf be greater than = 1 and less than = 3. This means that we can have a total of 3x3 factors of 1600 which 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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16 Jul 2019, 03:31
1
All factors of 1600=2^6*5^2

Total number of factors of 1600, including 1 and 1600 is (6+1)∗(2+1)=7∗3=21 total factors. 21 includes all odd and multiples of 16 such as 5^1, and 5^2, and 2^4 (16), and 2^5 (32), 2^6(64). Now, we can count factors that are not multiple of 16. Those are 2^1, and 2^1*5^1, and 2^1*5^2, and 2^2, and 2^2*5^1, and 2^2*5^2, and 2^3, and 2^3*5^1, 2^3*5^2, 9 factors, which is C.
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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16 Jul 2019, 04:23
1
Easy one. Factorising 1600, we get 1600 = 2^6 * 5^2

Now, we are given two conditions 1) the divisors must be even and 2) the divisors should not be multiples of 16

Conforming to condition 1), we need to have atleast one 2 for the divisor to be even and conforming to condition 2), we cannot have more than three 2s (four 2s is 2^4 (16) and would make the divisor a multiple of 16).

Thus, we have three allowed powers for 2, i.e. 0, 1 and 2; and three allowed powers for 5, i.e. 0, 1, 2

Multiplying both, 3*3 = we get 9 factors of 1600, in total, which would be even and would not be a multiple of 16.

Hence, (C) is the correct answer choice.
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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16 Jul 2019, 04:29
1
How many even divisors of 1600 are not multiples of 16?

1. Let's find prime factors of 1600 and 16:
1600: 2^(6)*5^(2)
16: 2^(4)

2. Next step, let's find a possible number of multiple of 16
Must be 2^(4)
Can be 2^(2)*5^(2): 2, 2 2, 2 5 ...
Quantity of variants: 3*3 = 9 ( 3 variant: 2 , 2 2 and no 2; 3 variant: 5, 5 5 and no 5)

3. Let's find the number of odd factors:
Can be:
1. odd: 1,5
2. odd = odd*odd: 5*5
no other variant, so the total number is 3

4. Let's find total number of factors:
2^(6)*5^(2)
#factors= 7*3 = 21

5. Subtract from the total number odd and multiple of 16

21-9-3 = 9

(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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16 Jul 2019, 04:35
B. 6.

Total number of divisors divisible by 16 (or, in other words, multiple of 16) = 9.
Out of these, odd divisors = 3.
So, even divisors = 9-3 = 6.
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Re: How many even divisors of 1600 are not multiples of 16?  [#permalink]

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16 Jul 2019, 04:49
1
My answer is C. I.e 9

1600
Divisors are from combination 2^6 *5^2

To be even divisors it's needs to have 2
Also if it's not a multiple of 16 then it shouldn't have 2^4
So following divisors of 1600 would not be multiple of 16
2
2*5
2*25
2^2
2^2*5
2^2*25
2^3
2^3*5
2^3*25

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

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16 Jul 2019, 05:26
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] 16 Jul 2019, 05:26

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