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Prime factor of last page – Pre RMO 2018 Problem 1 Discussion

A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the third. The number of pages in the second volume is 50 more than in the first volume, and the numberpages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is 1709. If n is the last page number. What is the largest prime factor of n?
Suppose the number of pages is the First Volume is t. It’s first page is numbered 1. Then the second volume has t+50 pages. Its first page is numbered t+1. Finally, the third volume has \( \frac{3}{2} \times ( t + 50) \) pages. Its first page is (t + t + 50 + 1 =) 2t + 51.
Sum of the page numbers of first pages of three volumes is: 1 + t +1 + 2t + 51 = 3t + 51. 3t + 51 = 1709 This implies t= 552
The total number of pages in three volumes is \( t + t + 50 + \frac{3}{2} \times (t+50) \\ = \frac {4t + 100 + 3t + 150}{2} \\ = \frac { 7 \times 552 + 250}{2} \\ = 2057 = 11^2 \times 17 \). Hence the greatest prime factor is 17.

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By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

2 replies on “Prime factor of last page – Pre RMO 2018 Problem 1 Discussion”

Didn’t get the second hint that how 1 + t + 1 + 2t + 51 = 3t + 51.

Why the two 1s re not added here? Why it isn’t 3t + 53?!

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