Arrange the following mathematical values related to factors, multiples, and primes in asc… — Armed Services Vocational Aptitude Battery (ASVAB)

Arrange the following mathematical values related to factors, multiples, and primes in ascending order (from least to greatest).

1The total number of positive factors of $120$
2The sum of the distinct prime factors of $330$
3The greatest common factor (GCF) of $84$ and $140$
4The least common multiple (LCM) of $2$ , $3$ , $4$ , and $5$

Answer

The correct ascending sequence is: the total number of positive factors of 120 (16), followed by the sum of the distinct prime factors of 330 (21), followed by the GCF of 84 and 140 (28), and finally the LCM of 2, 3, 4, and 5 (60).

Evaluating each statement yields the numerical values

16

21

28

, and

60

. Arranging these from least to greatest gives the correct order: Factor count of

120

(16), Prime sum of

330

(21), GCF of

84

and

140

(28), and LCM of

2, 3, 4,

and

5

(60).

Step-by-Step Solution

Calculate the total number of positive factors of

120

16

By finding the prime factorization (

120 = 2^3 \times 3^1 \times 5^1

) and adding

1

to each exponent before multiplying them, we get

(3+1) \times (1+1) \times (1+1) = 4 \times 2 \times 2 = 16

Calculate the sum of the distinct prime factors of

330

21

The prime factorization is

330 = 33 \times 10 = 3 \times 11 \times 2 \times 5

. The distinct primes are

2, 3, 5,

and

11

. Their sum is

2 + 3 + 5 + 11 = 21

Determine the greatest common factor (GCF) of

84

and

140

28

Using prime factorizations (

84 = 2^2 \times 3 \times 7

and

140 = 2^2 \times 5 \times 7

), the common prime factors are

2^2

and

7

. Multiplying these yields

4 \times 7 = 28

Determine the least common multiple (LCM) of

2, 3, 4,

and

5

60

The LCM must include the highest power of each prime factor present in the numbers:

2^2

(from

4

3^1

, and

5^1

4 \times 3 \times 5 = 60

Order the evaluated results from least to greatest.

16 < 21 < 28 < 60

Sorting the calculated values provides the final correct arrangement of the mathematical statements.

Key Concept

Applying Prime Factorization to Calculate Factor Counts, GCF, and LCM

Alternative Method

Instead of full prime factorization for the GCF, the Euclidean algorithm can be used:

140 - 84 = 56

. Next,

84 - 56 = 28

. Next,

56 - 28 = 28

. The GCF is

28

Estimated Time:2m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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