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# help

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A book with 50 pages numbered 1 through 50 has its pages renumbered in reverse, from 50 to 1. For how many pages do both sets of page numbers share the same ones digit?

Jan 6, 2019

#1
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For this question, I'm going to assume you are asking the number of sheets of paper (sides) in which the old page # (side) and the new page # (side) have the same units digit.

This problem might seem difficult, but a little bit of magic (algebra and logic) solves it.

Page 1 is replaced with 50, Page 2 is replaced with 49...ect. page 25 is replaced with page 26. Notice how all the old and new pages add up to 51.

51 is an odd number, and the only way to get an odd number through addition with two numbers is to have and odd number + even number. So this is algebraically and logically impossibe. For any integer A, A + A = 2A, which is always even. An odd number and an even number cannot have the same units digit, so the answer is \(\boxed{0}\) pages share the same ones digit.

Hope this helps,

- PM #1
+2

For this question, I'm going to assume you are asking the number of sheets of paper (sides) in which the old page # (side) and the new page # (side) have the same units digit.

This problem might seem difficult, but a little bit of magic (algebra and logic) solves it.

Page 1 is replaced with 50, Page 2 is replaced with 49...ect. page 25 is replaced with page 26. Notice how all the old and new pages add up to 51.

51 is an odd number, and the only way to get an odd number through addition with two numbers is to have and odd number + even number. So this is algebraically and logically impossibe. For any integer A, A + A = 2A, which is always even. An odd number and an even number cannot have the same units digit, so the answer is \(\boxed{0}\) pages share the same ones digit.

Hope this helps,

- PM #2
+1

Very well done, PM!!!!   Jan 6, 2019