﻿ Class 10 NCERT Math Solution
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TOPICS
Formulae
Euclid Theorem :-
```1. Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.< b
2. If p is a prime and p divides a², then p divides a, where a is a positive integer.
```
Factorization :-
```1. Let x be a rational number whose decimal expansion terminates.
Then we can express x in the form p/q , where p and q are coprime, and
the prime factorisation of q is of the form 2n5m,
where n, m are non-negative integers.
2. Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2m × 5n,
where n, m are non-negative integers.
Then x has a decimal expansion which terminates.
3. Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2m × 5n,
where n, m are non-negative integers.
Then x has a decimal expansion which is non-terminating repeating (recurring).
```
HCF & LCM :-
```1. for any two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.
2. for any two positive integers a and b, HCF(a,b) = (a × b) × LCM(a,b).
3. for any two positive integers a and b, LCM(a,b) = (a × b) x HCF(a,b).
4. HCF(p,q,r) × LCM(p,q,r) ≠ p × q × r, where p, q, r are positive integers.
However, the following results hold good for three numbers p, q and r :
i)  LCM(p,q,r) = [p.q.r.HCF(p,q,r)] ÷ [HCF(p,q).HCF(q,r).HCF(p,r)]
ii) HCM(p,q,r) = [p.q.r.LCM(p,q,r)] ÷ [LCM(p,q).LCM(q,r).LCM(p,r)]
```
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