Session 8: Perfect Squares | A NEW OUTLOOK – CSIR-NET Series

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Live Lecture

Warm Up Questions

Definitions

Perfect Square is a natural number that can be written as the product of two equal factors.,

For example: 4, 25, 81, 441, …

Observations

1. Last Digit of the Multiple is influenced only by the Last Digits of the Numbers.
2. No perfect square ends with 2, 3, 7, 8.
3. No perfect square ends with an odd number of zeros.
4. Upon prime factorization, all their prime factors have even multiplicities.

Question

Abhirup had a question on “Upon prime factorization, all their prime factors have even multiplicities.” How does that work?

Here’s how. For example, consider 36. 36 = 6² .

Prime factorisation of 36 is given by 36 = 2² x 3².

Another example will be 484 = 22² = 11² x 2².

Why is it so? We can give prime factorisation of any natural number, say n = a^m x bⁿ x c^p x …, where a, b, c, … are prime numbers.

When you square it, n² = (a^m)² x (bⁿ)² x (c^p)² x … = a^2m x b²ⁿ x c^2p x …

Thus, all the prime numbers have even multiplicities.

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