Question

Find the total number of factors of 5¹³ + 5¹² + 5¹¹ + 5¹⁰.
1. 132
2. 143
3. 46
4. 20

Answer 1

Correct Answer - Option 1 : 132

Given:

The number is 513 + 512 + 511 + 5¹⁰ 

Concept used:

If the number can be expressed as pa.qb.r^c.s^d, where p, q, r, and s are prime numbers and a, b, c and d are whole numbers,

Then total number of factors of the number = (a + 1) × (b + 1) × (c + 1) × (d + 1)

Calculation:

513 + 512 + 511 + 510 

⇒ 5¹⁰ × (5³ + 5² + 5 + 1)

⇒ 510 × (125 + 25 + 5 + 1)

⇒ 510 × 156

⇒ 510 × 2 × 2 × 3 × 13

⇒ 510 × 2² × 3¹ × 13¹

Total number of factors = (10 + 1) × (2 + 1) × (1 + 1) × (1 + 1)

⇒ Total number of factors = 11 × 3 × 2 × 2

∴ Total number of factors are 132