15.7Math Education · Grade 5
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Chapter 15 · Divisibility

Lesson 15.7 — Chapter 15 Exercise

A complete English student edition of Test 15 from this exercise. Record every final answer, show reasoning for the three extended problems, and use exercise review to check your work .

13 questions120 pointsWorked review availableAutosaves in this browser

Exercise directions

  1. Try the questions. Enter your answers and show your working where requested.
  2. Submit your attempt. Choose Submit exercise & open review when you are ready.
  3. Review and improve. Use the help available for this submission. Edit the original answer fields, then submit again.

Your work saves automatically in this browser. Use Print or Reset at the top of the page when needed.

Tips for this exercise

Enter final responsesQuestions 1–3 are multiple choice. Questions 4–10 need short final answers. Questions 11–13 need a final answer and written reasoning.
Use divisibility evidenceShow the exact tests you used: factors, LCM, last digits, digit sums, or a complete candidate search.

Section I — Multiple choice

Questions 1–3 · 6 points each · 18 points

18 points
1

GCD, LCM, and scaling

Not answered

The greatest common divisor (GCD) of two positive whole numbers is 20, and their least common multiple (LCM) is 100. How many of the following statements are correct?

  1. ① The product of the two numbers is 2,000.
  2. ② If both numbers are multiplied by 10, their GCD is multiplied by 10.
  3. ③ If both numbers are multiplied by 10, their LCM is multiplied by 10.
  4. ④ If both numbers are multiplied by 10, their product is multiplied by 100.
Correction to an earlier version of this page

Statement ② now says “multiplied by 10,” matching the worked example. The earlier page incorrectly said 100. Your saved choice has been retained; please reconsider it using the corrected statement. Teachers should account for the earlier wording when reviewing previous work.

Choose the number of correct statements.
Question 1 choice
Optional scratch work
2

Rearrange numbers under a divisibility condition

Not answered

Rearrange the seven numbers

39, 41, 44, 45, 47, 52, 55

in a row so that the sum of every three consecutive numbers is a multiple of 3. Among all such arrangements, what is the greatest possible value of the fourth number?

Choose one answer.
Optional arrangement or reasoning
3

Form multiples of 9 without repeating digits

Not answered

Using one or more of the digits

1, 3, 4, 7, 9

form positive integers without repeating a digit. How many of the resulting integers are multiples of 9?

Interpretation used by the worked exampleA number may use any nonempty selection of the listed digits, but no digit may be used more than once.
Choose one answer.
Optional list of candidates

Section II — Fill in the blanks

Questions 4–10 · 6 points each · 42 points

42 points
4

Greatest common multiple under 1,000

Not answered

What is the greatest three-digit number divisible by 4, 5, and 6?

three-digit number
Optional scratch work
5

Use an equation and divisibility by 9

Not answered

Here a and b are digits from 0 through 9; digit strings are numbers, not products. The three-digit number 2a3 plus 326 equals the three-digit number 5b9. If 5b9 is divisible by 9, find a+b.

2a3+326=5b9
sum
Optional scratch work
6

Complete the greatest multiple of 15

Not answered

Fill the two boxes in 3□2□ with suitable digits (the two digits may differ) so that the four-digit number is divisible by 15. What is the greatest possible four-digit number?

3□2□
four digits
Optional candidate search
7

Digits only 0 and 1

Not answered

In base ten, find the least positive integer whose digits are only 0 and 1 and that is divisible by 225.

Number conventionThe worked example says “least natural number.” This student edition uses the intended nonzero interpretation; otherwise 0 would make the task trivial. A positive integer may not begin with 0.
digits 0 or 1
Optional reasoning
8

Count three-digit multiples of both 3 and 5

Not answered

Choose any 3 of the digits

3, 5, 0, 1

to form a three-digit number with no repeated digit. How many such numbers are divisible by both 3 and 5?

numbers
Optional list of valid numbers
9

A repeated missing digit

Not answered

The seven-digit number 22A333A is divisible by 4. Its last two digits form the two-digit number 3A, which is a multiple of 6. Find A.

22A333A
digit
Optional scratch work
10

Smallest multiple using only 0 and 8

Not answered

A is a positive integer. It is a multiple of 15, and every digit of A is either 0 or 8. Find the smallest possible value of A.

positive integer
Optional reasoning

Section III — Extended response

Questions 11–13 · 20 points each · 60 points

60 points
11

Move the units digit, then add

Not answered

Four students completed the following addition exercise:

Start with any six-digit number. Move its units digit, which is not 0, to the left of the original leftmost digit to make a new six-digit number. Then add the new number to the original number.

The four answers were:

172,535 · 568,741 · 620,708 · 845,267

The students may have chosen different starting numbers. Exactly one reported sum can be obtained by this process. What is the correct answer?

Select the correct reported sum.
Question 11 answer
12

Recover the price from a partly missing receipt

Not answered

Teacher Li bought 28 pens of the same price for the school. The total cost was 9□.2□ yuan. The two boxes stand for the same digit. Each pen’s price is a whole number of fen (100 fen = 1 yuan, so 1 fen = 0.01 yuan). What was the price of each pen?

ItemPens
Quantity28
Total9□.2□ yuan
13

Append three digits to make the smallest common multiple

Not answered

Append three digits to 865 (each appended digit may be 0 through 9, and digits may repeat) to form a six-digit number. The completed number must be divisible by 3, 4, and 5. Find the smallest possible number.

865□□□
six digits