CBSE Class 10 Mathematics Real Numbers Notes Set 14

Divisibility, HCF and LCM

A number is divisible by 4 if its last two digits are, by 8 if its last three are, by 9 if its digit sum is, and by 11 if the alternating digit sum is a multiple of 11. HCF × LCM = product of the two numbers. The largest equal part into which several lengths can be cut is their HCF; the time at which periodic events coincide is the LCM of the periods.

Remainders and Cyclicity

Reduce each factor modulo the divisor and multiply. Unit digits of powers repeat in cycles (for 2, 3, 7, 8 the cycle length is 4; for 4 and 9 it is 2; for 0, 1, 5, 6 it is 1). xⁿ − yⁿ is divisible by x − y; for even n also by x + y; xⁿ + yⁿ is divisible by x + y for odd n.

Number Bases

To convert from base 10 to base b divide repeatedly by b and read the remainders upward. A number written dₙ...d₁d₀ in base b equals dₙbⁿ + ... + d₁b + d₀.

Logarithms and Surds

log(ab) = log a + log b, log(a/b) = log a − log b, log aⁿ = n log a, log_b a = 1/log_a b. The number of digits of N is the integer part of log₁₀ N plus one. To compare surds, bring them to a common index and compare the radicands.

Solved Examples

Example 1. Find the HCF and LCM of 12 and 18.
Answer:
12 = 2² × 3 and 18 = 2 × 3². HCF = 2 × 3 = 6, LCM = 2² × 3² = 36.
HCF 6, LCM 36

Example 2. Find the remainder when 7¹⁰⁰ is divided by 5.
Answer:
7 ≡ 2 (mod 5) and 2⁴ ≡ 1, so 7¹⁰⁰ ≡ 2¹⁰⁰ = (2⁴)²⁵ ≡ 1.
Remainder 1

Example 3. Find the unit digit of 3⁴⁵.
Answer:
The unit digits of powers of 3 repeat 3, 9, 7, 1. 45 = 4 × 11 + 1.
3

Example 4. Is 4752 divisible by 11?
Answer:
Alternating sum: 2 − 5 + 7 − 4 = 0, a multiple of 11.
Yes

Example 5. Convert 45 to base 2.
Answer:
45 = 32 + 8 + 4 + 1.
101101

Example 6. Find the value of log₂ 64 + log₃ 81.
Answer:
6 + 4.
10

Example 7. Simplify log 8 + log 125.
Answer:
log(8 × 125) = log 1000.
3

Example 8. Find the number of divisors of 360.
Answer:
360 = 2³ × 3² × 5, so the number of divisors is 4 × 3 × 2.
24

Example 9. Which is greater, the cube root of 2 or the square root of 3?
Answer:
Raise both to the power 6: 2² = 4 and 3³ = 27.
The square root of 3 is greater

Example 10. The HCF of two numbers is 4 and their LCM is 48. If one number is 12, find the other.
Answer:
Product = 4 × 48 = 192, so the other number = 192/12.
16