A communications technician is configuring signal transponders. The synchronization cycle … — Armed Services Vocational Aptitude Battery (ASVAB)

A communications technician is configuring signal transponders. The synchronization cycle for each transponder is determined by the Least Common Multiple (LCM) of its two input frequencies (measured in Hz). Arrange the following pairs of input frequencies in ascending order based on their resulting synchronization cycle (smallest LCM to largest LCM).

1Input frequencies of $12\text{ Hz}$ and $18\text{ Hz}$
2Input frequencies of $16\text{ Hz}$ and $24\text{ Hz}$
3Input frequencies of $10\text{ Hz}$ and $25\text{ Hz}$
4Input frequencies of $8\text{ Hz}$ and $14\text{ Hz}$
5Input frequencies of $15\text{ Hz}$ and $20\text{ Hz}$

Cevap

The correct order from smallest to largest LCM is: 12 Hz and 18 Hz (LCM=36), 16 Hz and 24 Hz (LCM=48), 10 Hz and 25 Hz (LCM=50), 8 Hz and 14 Hz (LCM=56), followed by 15 Hz and 20 Hz (LCM=60).

The correct sequence is determined by finding the LCM of each frequency pair and listing them in ascending order. The calculated LCMs are 36, 48, 50, 56, and 60 respectively.

Adım Adım Çözüm

Calculate the Least Common Multiple (LCM) for the first pair,

12\text{ Hz}

and

18\text{ Hz}

The LCM of

12

and

18

36

The multiples of

18

are

18, 36, 54

. Since

36

is divisible by

12

(

12 \times 3 = 36

), it is the lowest common multiple.

Calculate the LCM for

16\text{ Hz}

and

24\text{ Hz}

The LCM of

16

and

24

48

The multiples of

24

are

24, 48

. Since

48

is divisible by

16

(

16 \times 3 = 48

), it is the lowest common multiple.

Calculate the LCM for

10\text{ Hz}

and

25\text{ Hz}

The LCM of

10

and

25

50

The multiples of

25

are

25, 50

. Since

50

is divisible by

10

(

10 \times 5 = 50

), it is the lowest common multiple.

Calculate the LCM for

8\text{ Hz}

and

14\text{ Hz}

The LCM of

8

and

14

56

Using prime factorizations:

8 = 2^3

and

14 = 2 \times 7

. The LCM takes the highest powers:

2^3 \times 7 = 8 \times 7 = 56

Calculate the LCM for

15\text{ Hz}

and

20\text{ Hz}

The LCM of

15

and

20

60

Using prime factorizations:

15 = 3 \times 5

and

20 = 2^2 \times 5

. The LCM takes the highest powers:

2^2 \times 3 \times 5 = 60

Arrange the pairs in ascending order based on their calculated LCMs.

36 < 48 < 50 < 56 < 60

The stem requires ordering the frequencies from the smallest resulting synchronization cycle (smallest LCM) to the largest.

Anahtar Kavram

Calculating the Least Common Multiple (LCM) to synchronize multiple periodic events.

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