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Prove this colouring sequential problem

Source: MTRP 2018 Class 11-Short Answer Type Question: Problem 6 :-

February 17, 2021
Coloring

Problem Statement

Let d(n)d(n) be the number of divisors of n,n, where nn is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence {a1,a2,⋯ }\left\{a_{1}, a_{2}, \cdots\right\} if {d(a1),d(a2),⋯ }\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\} is an nonconstant geometric series then {a1,a2,⋯ }\left\{a_{1}, a_{2}, \cdots\right\} does not bear same colour.
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