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CHAPTERS 3 & 4 ASSESSMENT
1. Write the first fifteen counting numbers in base four.
1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33
2. Write the number preceding and succeeding each of the following.
(a) EET twelve
Before = EE9 (twelve)
After = EEE (twelve)
(b) 10011 two
Before = 10010 (two)
After = 10100 (two)
3. If 2b six = 17 eight, what is the value of b? _______________
17(eight) = 15 (ten)
15 (ten) = 23 (six)
So:
B=3
4. Identify the whole number property illustrated in each of the following.
(a) 4 + (7 + 3) = 4 + (3 + 7)
Commutative for addition
(b) 5 • 1 = 1 • 5
Commutative for multiplication
(c) 5 • (5 • 6) = (5 • 5) • 6
Associative for multiplication
(d) 4 • (5 + 6) = (4 • 5) + (4 • 6)
Distributive
5. Find the missing numbers in each of the following:
A:
574 – 326 = 248
B:
146 + 561 + 328 = 1035
6. Place the digits 2, 4, 5, 6, and 8 in the boxes to obtain
(a) the greatest difference
865-24=841
(b) the greatest product
652*84= 54768
7. Hugo’s checking account at the beginning of the month had a balance of $250.
During the month he wrote five checks for $15, two checks for $12, and one
check for $107. He made one deposit for $62. What is
his new balance?
250-5*15-2*12-107+62
= $106
8. For each of the following, find all integer values of x that make the equation or
inequality true.
(a) x2 = 9
square root:
-3 or 3
(b) │ x│ = 5
Absolute value:
X=5
Or –x = 5
5 or -5
(c) │x + 2│ = 7
X+2=7 so x = 5
Or –x-2 = 7 so x = -9
-9 or 5
(d) x + 7 = 34 - 2x
Add 2x:
3x+7=34
Subtract 7:
3x = 27
Divide by 3:
X=9
(e) (x - 3)2 = 64
Square root:
x-3 = +/- 8
Add 3:
X = 3+/-8
X = 11 or -5
9. Use the chip model to illustrate and explain why 3 + -2 = 3 - 2.
X is positive and Y is negative, so:
XXX + YY = XXX – XX = X
We can add an XY pair to the right side, without changing the value, since X and
Y cancel each other out:
XXY
And another:
XXXYY
That’s the left, so we proved it.
10. Factor each of the following expressions.
(a) 5x - 3x2
x(5-3x)
(b) 25 - x2
(5-x)(5+x)
11. Evaluate the following when x = -3, if possible.
(a) –x
-(-3) = 3
(b) │x│
|-3| = 3
(c) -x2
-(3^2) = -9
(d) -(1+ x)
-(1-3) = -(-2) = 2
12. Write the first six terms of each of the sequences whose nth term is
(a) ( -3) n
-3, 9, -27, 81, -243, 729
(b) 3 – 4n
-1, -5, -9, -13, -17, -21
13. Find the least whole number with exactly seven positive divisors and explain
why it is the least.
If there are an odd number of divisors, the number must be a square number.
Let’s find a pattern in the squares…
1 (1 div)
4 (3 div)
9 (3 div)
16 (5 div)
25 (3 div)
36 (8 div)
49 (3 div)
64 (7 div)
The answer is 64
14. Determine whether each of the following numbers is prime or composite.
(a) 219
Divisible by 3, so composite
(b) 791
Divisible by 7, so composite
(c) 1001
Divisible by 11, so composite
15. Find each of the following.
(a) GCD(12, 26, 65)
12=2*2*3
26=2*13
65=5*13
GCD = 1
(b) LCM(12, 26, 65)
= 2*2*3*13*5
= 780
3. If the cornerstone represents when a building was built and it reads MCMXXII, when
was this building built?
Start from the left:
M = 1000
CM = 100 before 1000, so 900
XX = 20
II = 2
Putting it together:
1922
A:
the 3 is the hundreds place
B:
The next to last 0 is the tens place
C:
The 7 is the thousands place
D:
The 8 is the hundred thousands place
a. EE0twelve
before = ETE(twelve)
after = EE1 (twelve)
b. 100000two
before = 11111(two)
after = 100001 (two)
c. 555six
before = 554(six)
after = 1000(six)
d. 100seven
before = 66(seven)
after = 101(seven)
e. 1000five
before = 444(five)
after = 1001(five)
f. 110two
before = 101(two)
after = 111(two)
3-2
5.) No
3-3
Let's first figure out what the machine does:
x+5
(x+5)*4
(x+5)*4-6
((x+5)*4 - 6) / 2
Simplify that last expression:
(4x + 20 - 6) / 2
(4x+14)/2
2x+7
So, we can fill in the table using that function.
2 | 11
4 | 15
0|7
6 | 19
12 | 31
Chapter 4
2
-13
8
-3(-3+2) = -3(-1) = 3
-3*-3 + -3*2 = 9 – 6 = 3
Divide by -2:
5x-3 = -13
Add 3:
5x = -10
Divide by 5:
x = -2
5
-1
-7
-10
11
-4
Set x= 0
2-0 = 2
Set x = 1 to see which direction this goes:
2-7 = -5
So: 2-7x > 2
Add the odd digits: 4+6+2 = 12
Add the evens: 3+8 = 11
Difference = 1, NOT divisible by 11
Is it divisible by 3?
2+4+2+8+0+0 = 16, which isn’t, so:
NO
The number 1 must be a divisor.
If 9 is a divisor, then so is 3
So now we have:
1, 2, 3, 5, 9
If it’s divisible by 2 and 3, then it’s divisible by 6. If it’s divisible by 2 and 5, then
it’s divisible by 10. If it’s divisible by 2 and 9 then it’s divisible by 18. If it’s
divisible by 3 and 5, then it’s divisible by 15. If it’s divisible by 3 and 10, then it’s
divisible by 30. If it’s divisible by 5 and 9, then it’s divisible by 45.
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Based on this, the number is 90.