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Write 2025 as 81 × 25, or as 45 × 45. You get the same primes both times. A prime is a number with exactly two factors.

The primes are 3, 3, 3, 3, 5, 5. This track is about why.

This is Chapter 1, Real Numbers, in your NCERT Maths book.

What you will learn

• Break into primes — divide until you reach 1.
• One set of primes — the same primes, however you split. Use it for HCF and LCM. HCF is the highest common factor. LCM is the lowest common multiple.
• Squares have even powers — every power in them is even. A power says how many times a prime appears.
• A prime that divides a square — it divides the number too.
• Proof for √2 and √3 — one plan, used twice.
• Proof for √5 — the same plan, in six lines.
• What a proof settles — a search tries some. A proof covers all.
• Write the proof in order — four moves, each using the one before.
• A number joined to a root — is 4 + √3 irrational?
• HCF and LCM of three numbers — the same primes, three rows.

Need a quick recap?

These earlier lessons help. Open any shaky one first.

• Primes inside — the primes that build a number.
• Showing it cannot be — a number that is not a fraction.
• The spiral — lengths like √2 on a number line.
• A decimal that stops — which fractions end.