Mock Test Maths 8 Chapter 5 “Number Play”

Figure It Out (Mock Test)

Name: ____________________________ Date: ___________ Class: ___________

Instructions: Answer each question. Show your work on the space provided or on a separate sheet.

1. The sum of five consecutive integers is 95. What are these integers?

2. Suppose q is the smallest of seven consecutive integers. Describe the other six integers in terms of q.

3. For each statement below, determine whether it is always true, sometimes true, or never true. Explain using algebra and give examples or counterexamples.
   (i) The sum of two odd numbers is a multiple of 4.

   (ii) If a number is not divisible by 12, then it is also not divisible by 6.

   (iii) If two numbers are not divisible by 8, then their sum is not divisible by 8.

   (iv) The difference between a multiple of 6 and a multiple of 9 is a multiple of 3.

   (v) The sum of a multiple of 4 and a multiple of 6 is always even.

4. Find a few numbers that leave remainder 1 when divided by 4 and remainder 2 when divided by 5. Give an algebraic description of all such numbers (for example, using congruences like x≡1(mod4)).

5. Riddle:
   "I have some marbles, not a huge heap,
   Grouped in 4s, two stay and peep.
   Grouped in 3s, one hides aside,
   Yet grouped in 7s they fit with pride.
   Less than 150 is my final plea,
   How many marbles are there — can you see?"

   Find the number of marbles.

6. Rahul writes several numbers that leave remainder 2 when divided by 4. He claims: “If you add any three such numbers, the sum will always be a multiple of 4.” Is Rahul correct? Explain.

7. Without calculating the full numbers, use remainders (algebraically and visually) to find the remainders when the following are divided by 9:
   Given 872≡8(mod9) and 1453≡4(mod9).
   (i) What remainder does 872+1453 leave when divided by 9?

   (ii) What remainder does 1453−872 leave when divided by 9?

8. Find the smallest positive integer that leaves remainders 3 when divided by 4, 2 when divided by 5, and 1 when divided by 6. Explain why it is the smallest.

Mock Test-2 : New "Figure it Out" Questions

1. The sum of three consecutive even numbers is 102. What are the numbers?
2. Suppose k−2 is the smallest of seven consecutive numbers. Describe the other six numbers in terms of k.
3. For each statement below, determine whether it is always true, sometimes true, or never true. Justify your claim using algebra and provide examples.
   • (i) The product of three consecutive integers is divisible by 6.
   • (ii) If a number is not divisible by 12, then it is also not divisible by 4.
   • (iii) The sum of a multiple of 4 and a multiple of 6 is a multiple of 24.
4. Find the smallest number greater than 50 that leaves a remainder of 1 when divided by both 4 and 7. Write an algebraic expression to describe all such numbers.
5. A box contains a number of marbles.
   • When grouped in 4s, 1 is left over.
   • When grouped in 5s, 1 is left over.
   • When grouped in 6s, 1 is left over.
   • When grouped in 7s, none are left over.
     What is the smallest possible number of marbles in the box?
6. A set of numbers all leave a remainder of 4 when divided by 8. A student claims, "If you add any two numbers from this set, the sum will always be a multiple of 8." Is the student's claim true?
7. When divided by 11, number A leaves a remainder of 9, and number B leaves a remainder of 5. Without calculating, what remainder will A×B leave when divided by 11?
8. Find a number that leaves a remainder of 2 when divided by 5, a remainder of 5 when divided by 8, and a remainder of 8 when divided by 11. What is the smallest such positive number? (Hint: Notice how the remainder relates to the divisor).

End of worksheet — good luck!