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

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

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

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

We can use two approaches to this problem. The first one is less time consuming but requires close familiarity with number properties. The second one is a manual method and thus more time consuming but still brings us to the correct answer. Memorizing the second is easier and thus helpful when we forget the first

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

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

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

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

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

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

\(1600 = 2^65^2\)

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

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.

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.

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

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

1600 = 2^6*5^2
Total nos. of divisors = (6+1)*(2+1) =21
Total odd divisors = 2+1 =3
Total even divisors = 21 -3 = 18

1600 = 16* (2^2 *5^2), Total divisors of 16 = 3*3 = 9
And, Req. = 18-9 =9.

So, I think C.

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