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Difficulty: MediumFactors, Multiples, and Prime Factorization

A cybersecurity protocol requires transmitting a data file of exactly 54005400 kilobytes. The system must divide this file into equally sized data packets without leaving any remaining data. For the encryption algorithm to function correctly, the size of each packet (in kilobytes) must be an integer that is a multiple of 1212. Based on these conditions, how many different packet sizes are possible?

  1. 18Answer
  2. B
    48
  3. C
    30
  4. D
    6

Answer

18
To find the number of factors of 54005400 that are multiples of 1212, we first divide 54005400 by 1212, which gives 450450. The number of possible packet sizes is exactly equal to the number of divisors of 450450. The prime factorization of 450450 is 21×32×522^1 \times 3^2 \times 5^2. Using the divisor-counting formula (adding 11 to each exponent and multiplying), we find there are (1+1)×(2+1)×(2+1)=2×3×3=18(1+1) \times (2+1) \times (2+1) = 2 \times 3 \times 3 = 18 valid packet sizes.

Step-by-Step Solution

1
Find the prime factorization of the total file size (54005400).
5400=54×100=23×33×525400 = 54 \times 100 = 2^3 \times 3^3 \times 5^2.
Prime factorization is required to find the total number of divisors.
2
Determine the mathematical condition for the packet size SS.
SS must be a divisor of 54005400 and a multiple of 1212 (22×312^2 \times 3^1). This means S=12×kS = 12 \times k.
Translates the real-world constraints into a mathematical relationship.
3
Find the maximum possible value and prime factorization of the multiplier kk.
kk must be a divisor of 540012=450\frac{5400}{12} = 450. The prime factorization of 450450 is 21×32×522^1 \times 3^2 \times 5^2.
The number of valid packet sizes SS is directly equal to the number of divisors of kk.
4
Calculate the total number of divisors of 450450.
(1+1)×(2+1)×(2+1)=2×3×3=18(1+1) \times (2+1) \times (2+1) = 2 \times 3 \times 3 = 18.
Applies the standard formula for counting factors (adding 1 to each prime exponent and multiplying them).

Key Concept

Finding the number of factors of a number that are multiples of another given number.
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Topics

Factors, Multiples, and Prime Factorization

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