CBSE Class 10 Mathematics Real Numbers Notes Set 15

Euclid's Division Lemma and HCF

For positive integers a and b there are unique integers q and r with \( a = bq + r \) and \( 0 \le r < b \). Repeating the division on b and r until the remainder is 0 gives the HCF as the last non-zero divisor.

Fundamental Theorem of Arithmetic

Every composite number is a product of primes in exactly one way, apart from order. Using this, HCF is the product of the smallest powers of common primes and LCM is the product of the greatest powers of all primes. For two numbers, \( HCF\times LCM = \text{product of the numbers} \). The number of factors of \( p^a q^b \) is \( (a+1)(b+1) \).

Remainders and Units Digits

A perfect square leaves remainder 0 or 1 on division by 3 or by 4. The units digit of powers cycles: 2, 4, 8, 6 for 2; 3, 9, 7, 1 for 3; always 6 for 6 and 5 for 5. A number is divisible by 3 or 9 when its digit sum is, and by 15 when divisible by both 3 and 5.

LCM and HCF of Fractions

\( LCM = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \) and \( HCF = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} \) when the fractions are in lowest terms.

Solved Examples

Example 1. Find the HCF and LCM of 96 and 404.
Answer:
\( 96 = 2^5\times3 \), \( 404 = 2^2\times101 \).
HCF = 4, LCM = 9696

Example 2. Use Euclid's algorithm to find the HCF of 135 and 225.
Answer:
\( 225 = 135\times1+90 \), \( 135 = 90\times1+45 \), \( 90 = 45\times2 \).
HCF = 45

Example 3. Find the greatest number that divides 70 and 125 leaving remainders 5 and 8.
Answer:
\( 70-5 = 65 \) and \( 125-8 = 117 \); HCF(65, 117) = 13.
13

Example 4. Find the units digit of \( 7^{51} \).
Answer:
The cycle of 7 is 7, 9, 3, 1 and \( 51 = 4\times12+3 \).
3

Example 5. How many factors does 360 have?
Answer:
\( 360 = 2^3\times3^2\times5 \).
\( (3+1)(2+1)(1+1) = 24 \). 24

Example 6. Find the LCM of \( \frac23 \) and \( \frac4{9} \).
Answer:
\( \frac{\text{LCM}(2,4)}{\text{HCF}(3,9)} = \frac43 \).
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