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Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips

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Bunuel
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Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips [#permalink] New post   22 Jul 2022, 01:30
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Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips the next number 2, marks out 3, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number 2, skips the next number 4, marks out 6, skips 8, marks out 10, and so on to the end. Josh continues in this manner until only one number remains. What is that number?

(A) 13

(B) 32

(C) 56

(D) 64

(E) 96


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chetan2u
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Re: Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips [#permalink] New post   22 Jul 2022, 20:07
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Bunuel wrote:
Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips the next number 2, marks out 3, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number 2, skips the next number 4, marks out 6, skips 8, marks out 10, and so on to the end. Josh continues in this manner until only one number remains. What is that number?

(A) 13

(B) 32

(C) 56

(D) 64

(E) 96


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Quick answer:

What are we cutting first?
The series starts with 1, and we cut all odd numbers?

What are we left with?
Multiples of 2.

What do we cut next?
The series starts with 2, and we cut Multiples of 2 but not 4

Thus series always starts with 2^n, and is left with multiples of 2^n. Thus the largest positive integer of type 2^n, which is less than 100 is our answer.
64 is 2^6 and our answer as 2^7 > 100.


You can do it step wise too.

I. 1,2,3,4,5…99,100
Cut out odd numbers

II. 2,4,6,…96,98,100
Cut out 2,6,10…, that is we are left with multiples of 4.

III. 4,8,12,…96,100
Cut out 4,12,20…, that is multiples of 4 but not 8

IV. 8,16,24,…96
Cut out 8,24,…, that is multiples of 8 but not 16

V. 16,32,…96
Cut out 16,48,80, that is multiples of 16 but not 32

VI. 32,64,96
Cut out 32 and 96

VII
64


D
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Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips [#permalink] New post   23 Jul 2022, 00:24
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From step 1 it says basically that you can eliminate all the odd numbers.

Step 2: you eliminate all the multiples of 2 which are not as the same time also multiples of 4

at the end of step 2 you will remain with only multiples of 4.

At this point you can write them down and cancel them out one by one following the procedures of step 1 and 2, You will end up with 64.


Answer D
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Re: Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips [#permalink] New post   25 Jul 2022, 18:30
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Bunuel wrote:
Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips the next number 2, marks out 3, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number 2, skips the next number 4, marks out 6, skips 8, marks out 10, and so on to the end. Josh continues in this manner until only one number remains. What is that number?

(A) 13

(B) 32

(C) 56

(D) 64

(E) 96



On the first pass, we eliminate everything that's not a multiple of 2.
On the second pass, we eliminate everything that's not a multiple of 4.
On the third pass, we eliminate everything that's not a multiple of 8.
...16.
...32.
...64.
So, we are left with 64.

Answer choice D.
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Re: Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips [#permalink]
25 Jul 2022, 18:30
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