341x,y,z ≡-1 mod 30
Show that there exist infinitely many positive integers A such that
(1) 2A is a perfect square ;
(2) 3A is a perfect cube ;
(3) 5A is a perfect fifth power.
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5 Answers
11We start with the fact that 2,3,5 | A.
So let A=2^x 3^y5^z
For nos like 2A, 3A, 5A and observing that (x+1), y, z are divisible by 2 & so are x,(y+1),z by 3, also 5| x,y,(z+1)
So let x=15+30k.
y=20+30k
z=24+30k.
Vary k over the set of naturals to get infinitely many such nos!
1@ prophet sir
are u refering to the x y z in soumiks solution...?and sir how did u get the value.....pls explain
341thats incorrect too. I too have slipped up.
The simplest way to write the values of x,y,z which Soumik has mentioned would be
x = 15 (2p-1)
y = 10 (3q-1)
z = 6 (5r-1)
