1. If x is the sum of six consecutive integers, then x is divisible by which of the following:
A.1
B.2
C.3
D.4
E.5
F.6
G.15
2. In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child
draws a card and multiplies the integer on the card by the next larger integer. If each possible product is
between 15 and 200, then the least and greatest integers on the cards could be
A.3 and 15
B. 3 and 20
C.4 and 13
D. 4 and 14
E. 5 and 15
3. The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of
these 3 integers?
A.17
B.16
C.15
D.14
E.13
4. If n is a positive integer, then what could be the units digit of 23n
A.0
B.1
C.2
D.4
E.6
F.8
G.9
5.What is the unit digit of 63n+5, n is a prime number.
6.If n is a multiple of 5 and n= p2q, where p and q are prime numbers, which of the following must be a
multiple of 25?
A.p2
B.q2
C.pq
D.p2q2
E..p2q3
7.What are the possible remainders when 'n',a positive integer, is divided by 15?
A.2
B.3
C.9
D.11
E.13
F.15
G.17
8.What is the sum of the first 10 prime numbers?
A.100
B.124
C.125
D.126
E.129
9. If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is
the least possible value of n?
A.2
B. 7
C.9
D.11
E.13
10. X is prime number,then what could value(s) of X if X is a factor of 15! ?
A. 5
B.8
C.9
D.11
E.12
F.15
G.17
11. 1727 has a units digit of:
12.Each digit in the two-digit number G is halved to form a new two-digit number H. Which of the following
could be the sum of G and H?
A. 152
B. 150
C. 124
D. 129
E. 138
13.11+22+33+...+1010 is divided by 5. What is the remainder?
A. 0
B. 1
C.2
D. 3
E. 4
14.What is the units digit of (71)5(46)3(103)4 + (57)(1088)3 ?
A. 0
B. 1
C. 2
D. 3
E. 4
15. What is the units digit of 17728 – 13323?
A. 1
B. 3
C. 4
D. 6
E. 9
16. What is the greatest integer m for which the number 50! / 10m is an integer?
A. 5
B. 8
C. 10
D. 11
E. 12
17.How many terminating zeroes does 200! have?
A. 40
B. 48
C. 49
D. 51
E. 54
18.When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by
9?
A. 0
B. 1
C. 2
D. 3
E. 6
19. When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19,
the remainder is also 3. What is the remainder when y is divided by 19?
A. 0
B. 3
C. 5
D. 11
E. 13
20.The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and
45 is 3. Which of the following could be the value of n?
A. 6
B. 8
C. 9
D. 12
E. 15
Explanation for Question1:
If x is the sum of six consecutive integers, then x is divisible by which of the following:
Answer – (C)
Solution:
Since x is the sum of six consecutive integers, it can be written as:
x = n + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5)
x = 6n + 15
In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child
draws a card and multiplies the integer on the card by the next larger integer. If each possible product is
between 15 and 200, then the least and greatest integers on the cards could be
Answer – (C)
Solution:
If the least number was 3, then 3*4=12<15, does not fulfill the requirement. So, the least number is 4.
If the greatest number is 14, then 14*15=210>200, does not fulfill the requirement.
So, answer is "4 and 13"
The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of
these 3 integers?
Answer – (A)
Solution:
12+52+72=75, so the sum of the three integers is (1+5+7)= 13.
Explanation for Question 4:
If n is a positive integer, then what could be the units digit of 23n
Answer-CDEF
Solution: when n=1, the units digit of 23=8
when n=2, the units digit of 26=4
when n=3, the units digit of 29=2
when n=4, the units digit of 23=6
5.What is the unit digit of 63n+5, n is a prime number.
Answer:6 Solution: 6positive integer=6
Explanation for Question 6:
If n is a multiple of 5 and n= p2q, where p and q are prime numbers, which of the following must be a
multiple of 25?
Answer – (A)Solution:p^2q is a multiple of 5, only can ensure that pq is a multiple of 5.
So, only (pq)^2 can surely be a multiple of 25.
Explanation for Question 7:
What are the possible remainders when 'n',a positive integer, is divided by 15?
Answer : the possible remainders when a nuber is divided by 15 are 0,1,2,....14.
Explanation for Question8:
What is the sum of the first 10 prime numbers?:
Answer: 2+3+5+7+11+13+17+19+23+29=129
Explanation for Question9:
If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is
the least possible value of n?
Answer-990=11*9*5*2, where 11 is a prime number. So, to guarantee that the produce will be a multiple of 990, the least possible value of n is 11
Explanation for Question 10: X is prime number,then what could value(s) of x if X is a factor of 15! Answer is 5,11. Solution: The prime numbers less than 'n' are all factors of n!.
Explanation for Question 11:
1727 has a units digit of:
Answer – 8 Solution: the units digit of 1727 is equal to 27 27=128.
Explanation for Question 12:
Each digit in the two-digit number G is halved to form a new two-digit number H. Which of the following
could be the sum of G and H?
. Solution: The answer must be multiple of three and it should be less than or equal to 147.
Explanation for Question 13:11+22+33+...+1010 is divided by 5. What is the remainder? Explanatory Answer The units digit of11=1 The units digit of22=4 The units digit of33=7 The units digit of44=6 The units digit of55=5 The units digit of66=6 The units digit of77=3 The units digit of88=6 The units digit of99=9 The units digit of1010=0 Sum of the units digit =47 The remainder when 47 is divided by 5 is 2.
Explanation for Question 14:What is the units digit of (71)5(46)3(103)4 + (57)(1088)3 Explanatory Answer The units digit of 715=7 The units digit of 463=6 The units digit of 1034=1 The units digit of(71)5(46)3(103)4 = 1 x 6 x 1=6---------> (i) The units digit of 10883=2 The units digit of (57)(1088)3= 7 x 2 =4---------->(ii) adding (i) and (ii) the answer is 0.
Explanation for Question 15 What is the units digit of 17728 – 13323?
Explanatory Answer
the units digit of 17728=1 the units digit of 13323=7 1-7=4 ( 17728 is greater than 13323)
Explanation for Question 16: What is the greatest integer m for which the number 50! / 10m is an integer?
[x] means integral part of x. eg. [3.9]=3,[4.2]=4 Here we have to find highest power of 10 in 50!, which means highest power of 5 in 50! Highest power of 5 in 50! = [50/5]+[50/52]+[50/53]= 10+2+0 =12.
Explanation for Question 17:
How many terminating zeroes does 200! have?
Explanatory Answer
[x] means integral part of x. eg. [3.9]=3,[4.2]=4 Here we have to find highest power of 10 in 200!, which means highest power of 5 in 200! Highest power of 5 in 200! = [200/5]+[200/52]+[200/53]+[200/54]= 40+8+1+0 =49.
Explanation for Question 18: When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by
9? Explaination: If x divided by 9 also has a remainder of 5, we can also express x as x = 9z + 5, where z is an integer. 3x=3(9z+5)=27z+15. The question asks us what the remainder is when 3x is divided by 9. i.e what the remainder is when 27z+15 is divided by 9. When 27z is divided by 9 the remainder is 0 When 15 is divided by 9 the remainder is 6. the answer is (0+6)=6.
Explanation for Question 19: When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by
9? If x divided by 11 has a quotient of y and a remainder of 3, x can be expressed as x = 11y + 3, where y is an integer (by definition, a quotient is an integer). If x divided by 19 also has a remainder of 3, we can also express x as x = 19z + 3, where z is an integer.
We can set the two equations equal to each other:
11y + 3 = 19z + 3
11y = 19z
The question asks us what the remainder is when y is divided by 19. From the equation we see that 11y is a multiple of 19 because z is an integer. y itself must be a multiple of 19 since 11, the coefficient of y, is not a multiple of 19.
If y is a multiple of 19, the remainder must be zero.
The correct answer is A
Explanation for Question 20: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and
45 is 3. Which of the following could be the value of n? Explaination: The greatest common factor of 16 and the positive integer n is 4, from this we can say n is a multiple of 4. the greatest common factor of n and4 5 is 3, from this we can say n is a multiple of 3. From the above two statements we can say n is multiple of 12. From the options we can say n is 12.