3rd tier government of India is not releated to Euclid's Division Lemma . 3rd tier is releated to political science whereas Euclid's Division Lemma is the part of mathematics.

The 3rd Tier of Government in India: Panchayati Raj System

The 3rd tier of government in India is known as the Panchayati Raj system. It is a decentralized form of governance that aims to involve local communities in decision-making processes. The term "Panchayati Raj" refers to the system of local self-government in rural areas. It was established to ensure effective and efficient administration at the grassroots level.

Key Features of the Panchayati Raj System:
- Constitutional Status: The Panchayati Raj system is provided constitutional status through the 73rd Amendment Act of 1992. This amendment added a new Part IX to the Constitution, which deals specifically with Panchayats.
- Three-Tier Structure: The system is organized into three tiers, namely the Gram Panchayat at the village level, the Panchayat Samiti at the block level, and the Zila Parishad at the district level. Each tier has its own set of powers and responsibilities.
- Elections: The members of the Panchayati Raj institutions are elected by the people through direct elections. The elections are held regularly, ensuring that the local communities have the opportunity to choose their representatives.
- Devolution of Powers: One of the main objectives of the Panchayati Raj system is to devolve powers to the local bodies. It empowers them to make decisions on various matters such as planning, implementation of development programs, and management of local resources.
- Financial Autonomy: The Panchayati Raj institutions have financial autonomy, which means they have the power to collect taxes, generate revenue, and receive funds from the state and central governments. This enables them to fund local development initiatives.
- Participation and Inclusion: The Panchayati Raj system aims to ensure participation and inclusion of all sections of society, including women and marginalized groups. It reserves seats for Scheduled Castes, Scheduled Tribes, and women in the Panchayati Raj institutions to promote their representation and participation in decision-making processes.

Euclid's Division Lemma (Optional)

Euclid's Division Lemma is a fundamental concept in number theory. It states that for any two positive integers "a" and "b," there exist unique integers "q" and "r" such that "a = bq + r," where "0 ≤ r < b."="" in="" simpler="" terms,="" it="" states="" that="" any="" positive="" integer="" "a"="" can="" be="" divided="" by="" another="" positive="" integer="" "b"="" and="" can="" be="" expressed="" as="" a="" quotient="" "q"="" multiplied="" by="" "b"="" plus="" a="" remainder="" "r"="" that="" is="" smaller="" than="" />

This lemma is often used as a basis for proving other theorems and properties in number theory. It allows us to understand the properties of division, factors, and multiples of integers. It also helps in solving problems related to divisibility and finding the greatest common divisor (GCD) of two numbers.

Euclid's Division Lemma is an optional topic in the Class 10 curriculum and may be covered depending on the syllabus and the discretion of the educational institution. It provides a foundational understanding of division and is a precursor to more advanced concepts in number theory.