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Question

Difficulty: MediumFactors, Multiples, and Prime Factorization

A numerical access code XX is generated based on the evaluation of the following mathematical expression:

X=36+144÷6×4X = 36 + 144 \div 6 \times 4

To successfully bypass the security protocol, a user must input the exact total number of positive factors (divisors) of the value XX. What number must the user input?

  1. A
    3
  2. B
    8
  3. 12Answer
  4. D
    16

Answer

The user must input 12, as the evaluated number 132 possesses exactly 12 positive factors.
To find the correct access code, the expression must first be properly evaluated using standard order of operations (BODMAS). By performing division and then multiplication left-to-right, we get 144÷6=24144 \div 6 = 24, and 24×4=9624 \times 4 = 96. Adding 36 yields X=132X = 132. The prime factorization of 132 is 22×31×1112^2 \times 3^1 \times 11^1. Using the standard formula for finding total divisors, we add 1 to each exponent and multiply them together: (2+1)(1+1)(1+1)=3×2×2=12(2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12.

Step-by-Step Solution

1
Evaluate the division and multiplication parts of the expression from left to right according to the BODMAS rule.
144÷6=24144 \div 6 = 24, followed by 24×4=9624 \times 4 = 96.
Division and multiplication hold equal precedence and must be executed left-to-right before any addition.
2
Complete the addition to find the final integer value of XX.
X=36+96=132X = 36 + 96 = 132.
Addition is performed after higher-precedence operations are resolved.
3
Determine the prime factorization of the computed value 132.
132=4×33=22×31×111132 = 4 \times 33 = 2^2 \times 3^1 \times 11^1.
Finding the prime bases and their respective powers is required to apply the divisor count formula.
4
Calculate the total number of positive factors by adding 1 to each exponent and multiplying the results.
(2+1)×(1+1)×(1+1)=3×2×2=12(2 + 1) \times (1 + 1) \times (1 + 1) = 3 \times 2 \times 2 = 12.
The formula (a+1)(b+1)...(a+1)(b+1)... systematically counts every possible product combination of the prime factors.

Key Concept

Factors, Multiples, and Prime Factorization
Estimated Time:1m 30s
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Topics

Factors, Multiples, and Prime Factorization

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