Prime Numbers
What is a prime number?
A number that only has 1 and itself as factors
7 + 7 / 7 + 7 =
15
Find the LCM of 12 and 36.
36
24 x 74 =?
Leave in simplified exponent form.
144
The sum of the first three prime numbers is:
10
Name all the prime factors of 90.
2, 3, 5
If a + b + c + d + e = 15, find the value of 3a + 3b + 3c + 3d + 3e.
45
Can the GCF of two numbers be 2.5? Explain.
No. Factors must be whole numbers.
What is a0?
(hint: a can be any number, so make sure to state any restrictions!)
1, a ≠ 0
These two numbers have the following prime factorizations:
23 x 52 x 7
22 x 3 x 5
Determine their LCM.
= 23 x 3 x 52 x 7
The sum of four different prime numbers is 31. Which number will always appear?
2
Let y = 1/x, where x is any real number. Is there any value for x that can't satisfy a value for y? Explain.
Yes - when x = 0, y = 1/0, which is undefined.
Without calculating, is 2156847 divisible by 7?
Yes
Express (1/3)(3-2)(1/9-4)(27) as a power of 3.
= 3-1 x 3-2 x 94 x 27
= 3-3 x 38 x 33
= 38
If 3a-2 = 4, solve for 3a+1.
108
Prime factorize 10!.
= 2 x 3 x (2 x 2) x 5 x (2 x 3) x 7 x (2 x 2 x 2) x (3 x 3) x (2 x 5)
= 28 x 34 x 52 x 7
Let a = 1 + 2 + 3 + 4 + 5 +...+ 100.
Let b = -1 + 2 - 3 + 4 - 5 ... - 99 + 100.
What is a+b?
= 4 + 8 + 12 +...+ 200
= 4 (1 + 2 + 3 +...+ 50)
= 4 (51)(25)
= 5100
Find the smallest integer that leaves a remainder of 1 when divided by 2, 3, 4, 5, 6, 7, or 8.
= 8 x 7 x 5 x 3 + 1
= 56 x 15 + 1
= 841
What is 1232 + 2(123)(77) + 772?
Hint: Does this look familiar to you?
40,000
Given that 15 + 25 + 35 + 45 + 55 + 65 = 12201, find the value of 65 + 95 + 125 + 155 + 185. You may use a calculator, but you are NOT allowed to directly evaluate the expression. You must show your work.
= (35)(25 + 35 + 45 + 55 + 65)
= (243)(12200)
= 2964600
The fastest way to determine if a number is prime is to divide it by every prime number up to the square root of that number. Explain why.
The square root acts as the "middle". Dividing by any number smaller than the square root will result in a number bigger than the square root. This way, you eliminate two possible factors at a time - one bigger and one smaller than the square root. There will be no need to divide by the bigger prime numbers.
How many trailing zeros does 12527 x 1621 have?
= 581 x 284
= 23 x 1081
It has 81 trailing zeros.
Find the divisibility rule for 64.
If the last 6 digits is a multiple of 64, then it is divisible by 64.
(This is very hard!) A circle centered at the origin (0, 0) in a coordinate plane is defined by the equation x2 + y2 = r2, where r is the radius of the circle and (x, y) being any point on the plane. Explain why this equation is always true for any point lying on the circle that is defined by this equation. You may have to draw this out to give a full explanation.
This question doesn't actually have a lot to do with circles; it actually has more to do with triangles.
The radius of the circle is always going to be of length r. If we pick any point (x, y) lying on the circle, we can draw a straight line that extends all the way to the x-axis, touching perpendicularly. Do the same for the point to the y-axis.
The two lines will have side lengths x and y, respectively. Apply the Pythagorean Theorem:
x2 + y2 = r2.
A room has 1500 switches. By default, they are off. 1500 students walk into the room, one at a time. The first student flips every single switch. The second person flips every second switch; the third person flips every third switch, and so on. In the end, how many switches are off?
38 switches are on because there are 38 perfect squares from 1 to 1500.
Therefore, 1462 switches are off.