**Question 1. Prove that √5 is irrational.**

**Solution :**Let us assume, to the contrary, that √5 is rational.

That is, we can find integers a and b (≠ 0) such that :

a | ||

√5 | = | |

b |

So, b √5 = a⋅

That is, we can find integers a and b (≠ 0) such that :

a | ||

√5 | = | |

b |

So, b √5 = a⋅

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

2 | | | 140 |

2 | | | 70 |

5 | | | 35 |

7 | | | 7 |

| | 1 |

∴ 140 = 2×2×5×7

= 2

- An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi.
- A lemma is a proven statement used for proving another statement.
**Euclid’s Division Lemma :**Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.**Euclid’s division algorithm :**Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers c and d, with c > d, is obtained as follows:

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