Number theory 7. Prove that for all positive integers n there exists a positive

Question

Number theory 7. Prove that for all positive integers n there exists a positive integer m and distinct positive integers a1, a2 an such that lcm(ai, a m for all 1 i j S n

Explanation / Answer

let n>0 m>0 a1, a2 ,a3, a4 are +ve integers now lcm of any two number is either direct multiplication of those number or it is a number whom both of number can divide with remainder 0. the LCM of two sets of numbers. Multiply each factor the greatest number of times it occurs in any of the numbers. one number 9 has two 3s, and another number 21 has one 7, so we multiply 3 two times, and 7 once. This gives us 63, the smallest number that can be divided evenly by 3, 9, and 21. so as your question is that for lcm of ai and aj   where i and j belong to (1,n). so lcm of these two must lie the range (max( ai and aj ) ,+ve infinity) and m being a +ve number it also belongs to  range (max( ai and aj ) ,+ve infinity) where m= lcm( ai, aj) so it is proved. now let take a example let  ai = 3 and aj = 4 now 3=31 , 4= 22   (2*2) so lcm = 3* 2*2 = 12. lcm( 3,4) = 12 as 12 can be divided by both 3 and 4. also 12 is a +ve number in range (max( 3and 4) ,+ve infinity). proved that m always exists.

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