How to resolve memmory exhausted error?

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I get the following error when running a script. I am sure that its a heavy script and i need it in my website.

Fatal error: Allowed memory size of 67108864 bytes exhausted (tried to allocate 16820 bytes)

I am using a free hosting service. Considering the extra 16820 bytes that i need for the script. Please advice

Can upgrading my membership to prime membership http://x10hosting.com/prime-membership resolve the problem?

Or should I go for an illuminated package.