1. Set I contains six consecutive integers. Set J contains all integers that result from adding 3 to each of the integers in set I and also contains all integers that result from subtracting 3 from each of the integers in set I. How man more integers are there in set J than in set I?
A. 0
B. 2
C. 3
D. 6
E. 9
2. What is the greatest of 5 consecutive intergers if the sum of these integers equals 185?
3. If S is the set of positive integers from 1-100 inclusive, which of the following is true?
A. S contains more 1-digit numbers than 2-digit numbers
B. S contains more numbers with units digit 0 than with units digit 9
C. S contains more multiples of 5 than multiples of 10
D. S contains more multiples of 8 than multiples of 2
E. S contains more even numbers than odd numbers
4. Which of the following number is divisible by 3 and 5, but not by 2?
A. 955
B. 975
C. 990
D. 995
E. 999
5. If m is an integer, which of the following could be true?
I. 17/m is an even integer
II. m/17 is an even integer
III. 17m is a prime number
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
6. How many primes less than 1000 are divisible by 7?
A. none
B. 1
C. more than 1 but less than 142
D. 142
E. more than 142
7. For the final step in a calculation, Paul accidentally divided by 1000 instead of multiplying by 1000. what should he do to his answer to correct it?
A. Multiply it by 1000
B. Multiply it by 100,000
C. Multiply it by 1,000,000
D. square it.
E. Double it.
simplify
8. |2(5)-8|=?
A. 2
B. -2
C. 18
D. -18
9. 3-|-8+4|
A.7
B.4
C.-7
D.-1
10. If A= {1,3,5,7,9,11} and B= {2,4,6,8,10,12} What is the intersection of A and B?
A. {}
B. {4,5,6}
C. {1,3,5,7,9,11}
D. {1,2,3,4,5,6,7,8,9,10,11,12}
11. P={0,1,2,3,4,5,6} and Q={2,4,6,8,10} What is the Union of P and Q?
#1. Just pick any 6 consecutive integers and try it out
#2. Consider the numbers
x-2, x-1, x, x+1, x+2
#3. Should be clear. Even if you're not sure about some of them, the right one jumps out.
#4. any multiple of 2 is even
#5. start by noting that 17 is prime
#6. trick question. what is a prime?
#7. hint: what is (nx)/(x/n) ?
#8. |10-8| ?
#9. what is |-8+4| ?
#10. all the numbers in either set
1. Set I contains six consecutive integers. Set J contains all integers that result from adding 3 to each of the integers in Set I and also contains all integers that result from subtracting 3 from each of the integers in Set I. How many more integers are there in Set J than in Set I?
To find the number of integers in Set J, we need to consider all the integers that result from adding 3 to each integer in Set I and subtracting 3 from each integer in Set I. Since Set I contains six consecutive integers, we can denote the first integer as "x." The six consecutive integers in Set I are then: x, x+1, x+2, x+3, x+4, and x+5.
To find the number of integers in Set J, we need to consider both the integers resulting from adding 3 and subtracting 3 from each integer in Set I. This means that for each integer in Set I, two additional integers are added to Set J.
Therefore, the number of integers in Set J is 2 times the number of integers in Set I. Since Set I contains 6 integers, Set J will contain 2 * 6 = 12 integers.
Thus, there are 12 - 6 = 6 more integers in Set J than in Set I.
Answer: D. 6
2. What is the greatest of 5 consecutive integers if the sum of these integers equals 185?
Let's denote the smallest integer in the set as "x." Since we are dealing with consecutive integers, the set will be: x, x+1, x+2, x+3, and x+4.
The sum of these integers can be expressed as:
x + (x+1) + (x+2) + (x+3) + (x+4) = 185
Combining like terms, we get:
5x + 10 = 185
Subtracting 10 from both sides, we have:
5x = 175
Dividing both sides by 5, we get:
x = 35
Therefore, the smallest integer in the set is 35.
The greatest integer will be: 35 + 4 = 39.
Answer: 39
3. If S is the set of positive integers from 1-100 inclusive, which of the following is true?
Let's analyze each statement:
A. S contains more 1-digit numbers than 2-digit numbers.
There are 9 one-digit numbers and 90 two-digit numbers in the set, so this statement is false.
B. S contains more numbers with units digit 0 than with units digit 9.
There are 10 numbers with units digit 0 and 11 numbers with units digit 9, so this statement is false.
C. S contains more multiples of 5 than multiples of 10.
There are 20 multiples of 5 and 10 multiples of 10 in the set, so this statement is true.
D. S contains more multiples of 8 than multiples of 2.
Both 8 and 2 are factors of 16, so there is an equal number of multiples of 8 and 2 in the set. This statement is false.
E. S contains more even numbers than odd numbers.
Since even numbers have the last digit of 0, 2, 4, 6, or 8, and odd numbers have the last digit of 1, 3, 5, 7, or 9, there are equal numbers of even and odd numbers in the set. This statement is false.
Therefore, the correct answer is:
Answer: C. S contains more multiples of 5 than multiples of 10.
4. Which of the following numbers is divisible by 3 and 5, but not by 2?
To be divisible by both 3 and 5, a number must end in 5 or 0 and have a sum of its digits divisible by 3.
Let's go through the given options:
A. 955: Not divisible by 5.
B. 975: Ends in 5 and the sum of its digits (9+7+5=21) is divisible by 3.
C. 990: Ends in 0 and the sum of its digits (9+9+0=18) is divisible by 3.
D. 995: Not divisible by 5.
E. 999: Not divisible by 5.
The only number that satisfies both conditions is 975.
Answer: B. 975
5. If m is an integer, which of the following could be true?
I. 17/m is an even integer.
For 17/m to be an even integer, m must be a factor of 17, which is a prime number. The only factors of 17 are 1 and 17.
II. m/17 is an even integer.
For m/17 to be an even integer, m must be divisible by 17. Any multiple of 17 will satisfy this condition.
III. 17m is a prime number.
If 17m is a prime number, m must be equal to 1. Otherwise, when m is a non-unit integer, 17 multiplied by m will not be a prime number.
Therefore, the possible conditions are:
I. False
II. True
III. False
Answer: B. II only
6. How many primes less than 1000 are divisible by 7?
To solve this, we need to find all the primes less than 1000 and see how many of them are divisible by 7.
Listing the primes one by one would be time-consuming, so we can use a rule to determine whether a number is prime or not. This rule states that if a number is not divisible by any prime number less than its square root, then it is prime.
The square root of 1000 is approximately 31.6, so we only need to check for divisibility by prime numbers up to 31.
Checking divisibility by 7 for numbers less than 1000, we find that 7, 14, 21, 28, ..., 994 are divisible by 7. Counting the multiples of 7, we get:
994 / 7 = 142
Therefore, there are 142 primes less than 1000 that are divisible by 7.
Answer: D. 142
7. For the final step in a calculation, Paul accidentally divided by 1000 instead of multiplying by 1000. What should he do to his answer to correct it?
Dividing by 1000 is equivalent to multiplying by 1/1000. To correct the mistake, Paul needs to multiply his answer by 1000.
Answer: A. Multiply it by 1000.
8. Simplify |2(5)-8|.
Evaluating the expression inside the absolute value first:
2(5) - 8 = 10 - 8 = 2
The absolute value of 2 is:
|2| = 2
Answer: A. 2
9. Simplify 3 - |-8+4|.
Evaluating the expression inside the absolute value first:
|-8+4| = |-4| = 4
Substituting this result back into the expression:
3 - 4 = -1
Answer: D. -1
10. If A = {1,3,5,7,9,11} and B = {2,4,6,8,10,12}, what is the intersection of A and B?
The intersection of two sets is the set of elements that are common to both sets.
Checking the common elements between A and B, we find that there are no elements that exist in both sets.
Therefore, the intersection of A and B is the empty set.
Answer: A. {}
11. P = {0,1,2,3,4,5,6} and Q = {2,4,6,8,10}. What is the union of P and Q?
The union of two sets is the set of all elements that are in either set.
Combining the elements from P and Q:
P = {0,1,2,3,4,5,6}
Q = {2,4,6,8,10}
The union of P and Q will be:
{0,1,2,3,4,5,6,8,10}
Answer: {0,1,2,3,4,5,6,8,10}