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The desert of IranNT2023

Source: Iran MO 3rd round 2023 , NT exam , P3

August 17, 2023

number theory

Problem Statement

Let

K

be an odd number st

S_2{(K)} = 2

and let

ab=K

where

a,b

are positive integers. Show that if

a,b>1

and

l,m >2

are positive integers st:

S_2{(a)} < l

and

S_2{(b)} < m

then :

K \leq 2^{lm-6} +1

(

S_2{(n)}

is the sum of digits of

n

written in base 2)

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