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 Magic 8's (Posted on 2006-09-18)
There is a number which has 8 divisors including, 8, itself, and 1. There is also a number which has 18 divisors, including 18, itself and 1. The difference between these numbers is 28. What are the two numbers?

 No Solution Yet Submitted by joshua Rating: 4.0000 (2 votes)

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 solutions | Comment 1 of 8

Since |A-B|=28 and A is divisible by 4, then B is also divisible by 4. Therefore: A=23p, B=2232q. Let's [A] is the number divisors of [A], so [A]=8, [B]=18. The number of divisors for any number is:

[P1k1P2k2....Pnkn]=(k1+1)*(k2+1)...(kn+1)

where Pn are prime factors.

Thus, [A]=4[p] and [B]=9[q] therefore p and q are primes (p>2 and q>3). So, if two primes, p>2 and q>3, satisfy |8p-36q|=28 or

|2p-9q|=7 (1)

`then 8p and 36q satisfy the condition of the problem. The first ten such prime pairs are:`
` `
`19 5`
`53 11`
`73 17`
`89 19`
`107 23`
`127 29`
`163 37`
`181 41`
`197 43`
`269 59`
` `
`So the first ten solutions are:`
` `
`152 180`
`424 396`
`584 612`
`712 684`
`856 828`
`1016 1044`
`1304 1332`
`1448 1476`
`1576 1548`
`2152 2124`
` `
`One can suggest that there are infinite number of solutions but I'm not going to prove it.<o:p></o:p>`

Edited on September 18, 2006, 3:26 pm

Edited on September 18, 2006, 3:30 pm
 Posted by Art M on 2006-09-18 15:21:19

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