Shortcut Trick
Number N = 22 × 31 × 52 × 71
For factors divisible by 50 (21 × 52) but not by 100 (22 × 52), the factor must contain exactly 21 and 52.
Required factors = (power of 2 is fixed at 1) × (power of 3 can be 0 or 1) × (power of 5 is fixed at 2) × (power of 7 can be 0 or 1)
Number of factors = 1 × (1 + 1) × 1 × (1 + 1) = 1 × 2 × 1 × 2 = 4
∴ The correct answer is 4.
Alternate Method
Given: N = 22 × 31 × 52 × 71
Formula: If N = ap × bq × cr, total factors = (p+1)(q+1)(r+1).
⇒ Divisible by 50 means the factor must have at least 21 and 52.
⇒ Not divisible by 100 means the factor cannot have 22 or higher (if available) along with 52.
⇒ In N = 22 × 31 × 52 × 71, the exponent of 2 must be exactly 1.
⇒ The exponent of 5 must be exactly 2 (as 52 is the maximum available and minimum required).
⇒ The exponents of 3 and 7 can be anything from 0 to their respective maximums.
⇒ Possible exponents for 2: {1} → 1 choice
⇒ Possible exponents for 5: {2} → 1 choice
⇒ Possible exponents for 3: {0, 1} → 2 choices
⇒ Possible exponents for 7: {0, 1} → 2 choices
⇒ Total factors = 1 × 1 × 2 × 2 = 4
∴ The correct answer is 4.
Additional Information
Total Number of Factors
For a number N = pa × qb × rc, total factors = (a+1)(b+1)(c+1).
Factors Divisible by a Number k
Divide N by k first: N/k = M. The number of factors of N divisible by k is equal to the total number of factors of M.
Prime Factorization
Always express the number in the form of powers of prime numbers (2, 3, 5, 7, etc.) before applying factor formulas.