The numbers between 50 and 60 which have 8 factors are 54and56.

• Factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
• Factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

Explanation

Let x be the number we want to find, and let’s assume that “between 50 to 60“ means 50 < x < 60, so 50 and 60 are not included in this range.

We can ”guess“ the number of factors of an integer using its prime factors. To have 8 or more factors, such number must have at least 3 distinct prime factors, or 2 prime factors, one of them is in the form of a³.

Example:
Prime factorization of 30 is 2 × 3 × 5. Its factors are 1, 2, 3, 5, 6, 10, 15, 30, so there are 8 factors of 30.

In finding the requested numbers, we will skip 53 and 59, because they are prime numbers.

• Prime factorization of 51 is 3 × 17.
  ⇒ 2 prime factors, none of them is in the form of a³. Skip!

• Prime factorization of 52 is 2² × 13.
  ⇒ 2 prime factors, none of them is in the form of a³. Skip!

• Prime factorization of 54 is 2 × 3³.
  ⇒ 2 prime factors, one of them is in the form of a³.
  ⇒ Investigate: Factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
  ⇒ 8 factors.
  ⇒ First solution.

• Prime factorization of 55 is 5 × 11.
  ⇒ 2 prime factors, none of them is in the form of a³. Skip!

• Prime factorization of 56 is 2³ × 7.
  ⇒ 2 prime factors, one of them is in the form of a³.
  ⇒ Investigate: Factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.
  ⇒ Second solution.

• Prime factorization of 57 is 3 × 19.
  ⇒ 2 prime factors, none of them is in the form of a³. Skip!

• Prime factorization of 58 is 2 × 29.
  ⇒ 2 prime factors, none of them is in the form of a³. Skip!

∴  Solutions are 54and56.

CONLUSION

∴  The numbers between 50 and 60 which have 8 factors are 54 and 56.

• Factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
• Factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

(We’ve found 2 numbers instead of one.)