1. Find the number of factors of 500.
Sol:
500 = 5 × 10 × 10 = 2² × 5³
∴ Number of factors = (2 + 1)(3 + 1) = 12 Answer
2. Find the number of even factors of 800.
Sol:
800 = 8 × 10 × 10 = 8 × 2 × 5 × 2 × 5 = 2⁵ × 5²
3. Find the number of odd factors of 800.
Sol:
800 = 2⁵ × 5²
4. Find the number of factors, odd factors and even factors of 1240.
Sol:
1240 = 124 × 10 = 31 × 4 × 10 = 2³ × 5 × 31
total number of factors ⟹ (3 + 1)(1 + 1)(1 + 1) = 16 Answer
5. Find the total number fo factors, odd factors and even factors of 2⁵ × 3² × 5³ × 7² × 11
Sol:
Total number of factors = (5 + 1)(2 + 1)(3 + 1)(2 + 1)(1 + 1)
= 432 Answer
= 5 × (2 + 1)(3 + 1)(2 + 1)(1 + 1)
= 360 Answer
Number of odd factors ⟹ 3² × 5³ × 7² × 11¹
= (2 + 1)(3 + 1)(2 + 1)(1 + 1)
= 72 Answer
6. Find the number of factors of 2⁵ × 3² × 5³ × 7² × 11 that are divisible by 10, divisible by 100, divisible by 1000.
Sol:
Factors that are divisible by 10 ⟹
= 5 × (2 + 1) × 3 × (2 + 1) × (1 + 1)
= 270 Answer
Factors that are divisible by 100 ⟹
= 4 × (2 + 1) × 2 × (2 + 1) × (1 + 1)
= 144 Answer
Factors that are divisible by 1000 ⟹
1000 = 2³ × 5³
= 3 × (2 + 1) × 1 × (2 + 1) × (1 + 1)
= 54 Answer
7. Given a number
2⁵×3²×5³×7²×11
(i) Find the sum of all factors
(ii) Find the sum of even factors
(iii) Find the sum of odd factors.
Sol:-
(i) Sum of all factors ⟹
(2⁰ + 2¹ + 2² + 2³ +2⁴ +2⁵)(3⁰ + 3¹ + 3²)(5⁰ + 5¹ + 5² + 5³)(7⁰ + 7¹ + 7²)(11⁰ + 11¹)
=(63)(13)(156)(57)(12)
= 87390576 Answer
8. Find the number of prime factors of:
6⁶×8⁸×10¹⁰×12¹²×14¹⁴
Sol:- Given expression can be re-written as:
(2×3)⁶×(2)⁸×(2×5)¹×(2×3)¹²×(2×7)¹⁴
= 2⁷⁸×3¹⁸×5¹⁰×7¹⁴
Just add all powers as it is to get number of prime factors which in this case is:-
78 + 1 8+10 + 14
= 120 Answer
9. Given a number:
1480
(i) Find number of all factors
(ii) Find the number of even factors
(iii) Find number of odd factors
(iv) Find number of prime factors
(v) Sum of all factors
(vi) Sum of even factors
(vii) Sum of odd factors
Sol:- 1480=148×10 = 4×37×10 =2³×5×37
∴ number of all factors = 4×2×2
= 16 Answer
∴ number of even factors = 3×2×2 = 12 Answer
∴ number of odd factors 2×2 = 4 Answer
(iv) 2³×5¹×37¹
∴ Number of prime factors = 3 + 1 + 1 = 5 Answer
(v) (2⁰ + 2¹ + 2²+ 2³)(5⁰ + 5¹)(37⁰ + 37¹)
= (1 + 2 + 4 + 8)(1 + 5)(1 + 37 )
= 3420 ⟵ Sum of all factors Answer
(vi) Sum of even factors =
(2¹ + 2² + 2³)(5⁰ + 5¹)(37⁰+ 37¹)
= (14)(6)(38)
= 3192 Answer
(vii) Sum of odd factors
= (5⁰ + 5¹)(37⁰ + 37¹)
= 228 Answer
10. Find the number of factors of 24³ – 16³ – 8³
Sol:
8(3³ – 2³ – 1)
= 8³ × 18
= 8³ × 2 × 3²
= 11 × 3
= 33 Answer