How to make this change without changing the core? [duplicate]

As you see there is the apply_filters() call, with the $tag parameter, which is the name of the hook, set to widget_tag_cloud_args. As starting point for getting into hooks, filters and actions I’d suggest you read the codex page about the Plugin API. You can use the hook to change the arguments, e.g. like this:

add_filter( 'widget_tag_cloud_args', 'wspe160041_change_tag_cloud_number_arg' );
function wspe160041_change_tag_cloud_number_arg( $args ) {
    $args[ 'number' ] = 10;
    return $args;
}