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Modems don't use PAT or NAT. Modems don't have that capability. Using modem would lead to assigning public IP address to device. Routers are NAT capable. If we are talking about IPv4, typically, broadband routers that are leased from ISP to users are configured for PAT. Routers can have integrated modem (e.g. xDSL routers) or modem can be added to it (as line card).<br />
With IPv6 there should be no NAT. NAT and tunneling techniques (6to4, NAT64, NAT PT, etc...) are only present in the case that IPv4 is in use in some portion of network and IPv6 in in use in other portion of network.
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Modems don't use PAT or NAT. Modems don't have that capability. Using modem would lead to assigning public IP address to device. Routers are NAT capable. If we are talking about IPv4, typically, broadband routers that are leased from ISP to users are configured for PAT. Routers can have integrated modem (e.g. xDSL routers) or modem can be added to it (as line card).
With IPv6 there should be no NAT. NAT and tunneling techniques (6to4, NAT64, NAT PT, etc...) are only present in the case that IPv4 is in use in some portion of network and IPv6 in in use in other portion of network.

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In old modem when we share the Connection by ICS (Internet Connection sharing) then which Method modem was used?
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In old modem when we share the Connection by ICS (Internet Connection sharing) then which Method modem was used?

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What technique old modem were used. PAT or NAT? and Now a days which technique broadband routers are used?
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What technique old modem were used. PAT or NAT? and Now a days which technique broadband routers are used?

PAT and NAT

Modem

A router is a networking device that forwards data packets between computer networks. Routers perform the "traffic directing" functions on the Internet. The most familiar type of routers are home and small office cable or DSL routers that simply pass data, such as web pages, email, IM, and videos between computers and the Internet. More sophisticated routers, such as enterprise routers, connect large business or ISP networks up to the powerful core routers that forward data at high speed along the optical fiber lines of the Internet backbone. Though routers are typically dedicated hardware devices, use of software-based routers has grown increasingly common.

Routers

The term broadband commonly refers to high-speed Internet access that is always on and faster than the traditional dial-up access. Broadband includes several high-speed transmission technologies such as digital subscriber line (DSL), cable, fiber optics, wireless (for which there is a separate topic), satellite and broadband over powerlines (BPL).

Broadband

Cisco PIX is a dedicated hardware firewall appliance; the Cisco Adaptive Security Appliance (ASA) is a firewall and anti-malware security appliance that provides unified threat management and protection the PIX does not. Other Cisco devices and systems include routers, switches, storage networking, wireless and the software and hardware for PIX Firewall Manager (PFM), PIX Device Manager (PDM) and Adaptive Security Device Manager (ASDM).