Student Voices: Factors, Multiples, and Survey Plans

Use factors and multiples to organize a school newspaper’s opinion survey.

Organize the Responses

Find greatest common factors to organize survey responses. Use every response when a problem says there are no leftovers.

The newspaper received 42 responses from sixth graders and 70 from seventh graders.

What is the greatest common factor of 42 and 70?

Two survey questions received 81 and 54 responses. What is the greatest common factor of these counts?

A. 9B. 18C. 27D. 54

Editors will divide these response cards among the greatest possible number of identical review packets. Each packet must contain the same number of cards from each topic, with no cards left over.

Opinion Survey Cards
Topic	Number of cards
Lunch choices	48
Club choices	64

What is the greatest number of packets they can make? packets

The editors have 35 paper responses and 64 online responses. Which statement correctly describes the greatest common factor of 35 and 64?

A. The greatest common factor is 1.B. The greatest common factor is 5.C. The greatest common factor is 7.D. The greatest common factor is 35.

Nia has 90 responses about school lunches and 75 about school clubs. She makes the greatest possible number of identical packets. Each packet has the same number of lunch responses and the same number of club responses, with none left over.

How many packets can she make?
How many lunch responses go in each packet?
How many club responses go in each packet?

Coordinate Survey Schedules

Find least common multiples. For schedules, find the first time after the start that both events happen together.

One editor checks incoming opinions every 8 minutes. Another checks every 12 minutes. Both check at the start.

In how many minutes will they next check together? minutes

The newspaper posts a lunch poll every 6 days and a club poll every 9 days. Both are posted today. In how many days will both next be posted on the same day?

A. 3 daysB. 15 daysC. 54 daysD. 18 days

The editors want the same number of responses in two separate opinion polls. One poll’s responses will be arranged in groups of 4. The other poll’s responses will be arranged in groups of 10. Neither poll can have an incomplete group.

What is the least positive number of responses each poll could have? responses

Sam checks the survey inbox every 3 minutes. Imani checks it every 12 minutes. Both check now. Which statement gives their next shared check time?

A. They will next check together in 3 minutes.B. They will next check together in 12 minutes.C. They will next check together in 15 minutes.D. They will next check together in 36 minutes.

Rewrite Survey Totals

Use the distributive property to factor each sum. Use the greatest common factor as the outside factor.

10.

A survey received 36 votes for longer lunch periods and 60 votes for more lunch choices. Factor the total using the greatest common factor.

36 + 60 = × ( + 5)

11.

The newspaper received 28 opinions on homework and 42 on class projects. Which expression factors 28 + 42 using the greatest common factor?

A. 7 × (4 + 6)B. 14 × (2 + 42)C. 14 × (2 + 3)D. 14 × (2 + 2)

12.

Two survey pages collected 63 and 84 responses. Complete the factored sum using the greatest common factor. Keep the terms in their original order.

63 + 84 = × ( + )

13.

There are 32 responses supporting a quiet lunch area and 56 supporting outdoor seating. The editors divide them into the greatest possible number of identical packets, with no leftovers. Which expression shows the number of packets multiplied by the responses of each type in one packet?

A. 8 × (4 + 7)B. 4 × (8 + 14)C. 8 × (4 + 56)D. 16 × (2 + 7)

14.

Luis rewrites a survey total as 24 + 40 = 4 × (6 + 10). He says he used the greatest common factor. Is his equation correct? Is his claim about the greatest common factor correct? Explain and rewrite the sum using the greatest common factor.