Questions for CMAT
SECTION II
QUANTITATIVE
ABILITY
CHAPTER
Integers
General
Information
Integers
Information
Integers
(Z): The set of positive numbers, and negative numbers including 0 are the
integers. It is denoted by Z or I. For examples:
Z
= {……… –3, –2, –1, 0, 1, 2, 3, …………}
Note:
i. The set of positive integers Z or I = {1, 2, 3, 4…………….}
ii. The set of negative integers Z or
I = {…………, –3, –2, –1}
iii. 0 is neither a positive integer
nor a negative integer but it is an even integer.
p
= qz + r means i.e. when an integer
"P" is divided by "q" , then, z is the quotient and
"r" is the remainder.
A
number "n" is the even if the remainder is zero when "n" is
divided by 2 i.e. n = 2z + 0.
Consecutive
Integers: n, n + 1, n + 2, …………….
Consecutive
odd/even integers: n + 2, n + 4, n + 6,……………
Exercise
1.
Suppose p is even and q is odd, which of the following cannot be an integer?
i.
a. (i) only b.
(ii) only c. (iii) only d. (i) & (iii) only
For a fractional expression to be an
integer, the denominator must divide evenly into the numerator.
Now, Statement I cannot be an integer. Since q is odd and p is even, p+q is odd.
Further, since p + q is odd, it cannot be
divided evenly by the even number p.
Hence,
For example, if p = 2 and q = 3, then
Finally, Statement III cannot be an integer. p2 = p⋅p is even since it is the product of two even numbers.
Further, since q is odd, it cannot be
divided evenly by the even integer p2. The answer is (E).
2.
If p, q and r are consecutive integers p < q < r, which of the following
must be true?
i. q - r = 1 ii.
a. (i) only b.
(ii) only c. (iii) only d. (i) & (iii) only
3.
When "m" is divided by 2, the remainder is 1 and when n is divided by
4, the remainder is 3, which of the following must be true?
a. "m" is even b. "n" is even c. "m + n" is even d. "mn" is even
4.
If x is both the cube and the square of an integer and x is between 2 and 200,
the value of x is:
a. 8 b.
16 c.
64 d.
80
5.
When "p" is divided by 2, the remainder is 1 and if q is divided by
6, the remainder is 1, which of the following must be true?
a. pq
+ 1 is even b.
6.
p and q are integers. If p is divided by 9 leaves a remainder is 1, which of
the following must be true?
i. p
is even ii. p is odd iii. p = 3z + 1 for some integer 'z'
a. (i)
only b. (ii) only c. (iii) only d. (i) & (iii) only
7.
Which one of the following numbers is the greatest positive integer x such that
3x is a factor of 275?
a. 5 b. 8 c. 10 d.
15
8.
A number when divided by 12 gives a remainder 7. If the same number is divided
by 6, then the remainder must be
a. 1 b. 2 c. 3 d.
4
9.
Let a, b and c be three integers and let 'a' be a perfect square. If , then,
which one of the following statements must be true?
a. c must be an even integer b. c must be an odd integer
c. c must not be a perfect square. d. c must be a perfect square.
10.
If two non-zero positive integers p and q are such that p = 4q, p <8, q
is
a. 1 b.
2 c. 3 d. 4
11.
If "n" is an integer, then, which one of the following expression
must be even?
a. n2 + 1 b. n(n + 1) c.
n(n + 1) d. n(n + 4)
If each of the
dimensions of a rectangle is increased 100%, the area is increased
a. 100% b.
200% c. 300% d. 400%
If each
of the dimensions is doubled, the area of the new rectangle is four times the
size of the original one. The increase is three times, or 300%.
If p is an even integer and q is an
odd integer, which of the following must be an odd integer?
A. p/q
B. pq
C. 2p+q
D. 2(p+q)
E. 3p/q
The GMAT - The Definitive Shortcuts
Guide is not only just a collection of simple tricks or shortcuts for the
quantitative reasoning portion of the GMAT. I strongly recommend studying for a
more complete and robust quant theory knowledge for the GMAT.
Number System Shortcuts
Algebra Shortcuts
Geometry Shortcuts
1.
Method to multiply 2-digit number
AB×CD=AC[AD+BC]BDAB×CD=AC[AD+BC]BD
63∗46=24[36+12]1863∗46=24[36+12]18
Add the middle term = 24 [48] 18
Keep first term intact, form the middle term
by adding 2 numbers also keep the last term same, which means 2(4+4)(8+1)8=28982(4+4)(8+1)8=2898
1. A's income is 25% more than B. By how much percent is B's income less
than A?
a. 20% b.
25% c. 16.66% d. 22.5%
Solution
The correct option is C 20%
Let B's income be Rs. 100
Then, A's income = Rs. 125
∴ Difference in income =
Rs. 25
∴ Percentage by which B's
income is less than A =
Hence, B's income is less than that of A by 20%.
The income of A is 20% higher than that
of B.
The income of B is 25% less than that
of C.
What percent less is A's
income from C's
income?
a.
7% b. 8% c. 10% d. 12.5%
Solution
The correct option is B 10%
We take B's
income = Rs. 100.
Then A's income = Rs. (100 + 20) = Rs. 120 by the given
condition.
Also B's income is 25% less than that
of C.
i.e B's income =(100−25)%=75% of C's income.
∴ C's
income = Rs.
∴ the income of A is less than that
of C
= Rs.
So the p.c of the shortage in A's income from C's income
=
Ans - Option C.
Akshay's income is 20% less than that of
Ajay. What percent is Ajay's income more than that of Akshay?
Solution
Given: Akshay's income is 20 less than Ajay's.
Income (Akshay′s) = Income
(Ajay′s) − 20% Income (Ajay′s)
I (Akshay′s) = I(Ajay′s)
× (1 −
I(Ajay′s)=
I(Ajay′s)=I(Akshay′s)
+
I(Ajay′s) =
I(Ajay′s) = I(Akshay′s)
+
I(Ajay′s) = I(Akshay′s)
+ 25% I (Akshay′s)
Therefore Ajay's income is 25% more than Akshay's.
A's income is 20% less than that of B; by what percent is
B's income more than that of A's?
Hint: Start this question by assuming the
income of B as x. Then calculate the income of A in terms of B, according to
the above condition given. Now, calculate how much the value of B's income is
more than A and then divide by the value of A's income to get the answer.
I hope the hint is sufficient to solve the question; however
if you find difficulty solving the question, you can refer to the solution
below.
Complete step-by-step answer:
Now, let us take the income of B as x.
⇒B=x⇒B=x
Now, the income of a is 20% less than B. Therefore, let us
calculate it
⇒A=x−20%x⇒A=x−20%x
⇒A=x−20100x⇒A=x−20100x
⇒A=45x⇒A=45x
Now, we have both A's and B's income.
To calculate by how much B's income is more than A's, let us
subtract the income of A from B as shown below,
⇒B−A=x−45x⇒B−A=x−45x
⇒B−A=15x⇒B−A=15x
Now, to calculate the difference in percentage, let us divide
the difference by A's income to get the fraction shown below,
⇒B−AA=15x45x⇒B−AA=15x45x
⇒B−AA=14⇒B−AA=14
Now, multiply the fraction by 100 to get the percentage by
which B's income is more than A.
⇒B−AA×100%=14×100%⇒B−AA×100%=14×100%
⇒B−AA×100%=25%⇒B−AA×100%=25%
Thus, the required answer is
25%25%
.
So, the correct answer is “
25%25%
”.
Note: In mathematics, a percentage is a
number or ratio expressed as a fraction of 100. It is often denoted using the
percent sign, "%", A percentage is a dimensionless number; it has no
unit of measurement. It does not have any units as it is a ratio of quantities
with the same unit multiplied by 100.
2.
Some properties of square and square root
(a) Complete square of a no. is possible if
its last digit is 0, 1, 4, 5, 6 & 9. If last digit of a no. is 2, 3, 7, 8
then complete square root of this no. is not possible.
(b) If last digit of a no. is 1, then last
digit of its complete square root is either 1 or 9.
(c) If last digit of a no. is 4, then last
digit of its complete square root is either 2 or 8.
(d) If last digit of a no. is 5 or 0, then
last digit of its complete square root is either 5 or 0.
(e) If last digit of a no. is 6, then last
digit of its complete square root is either 4 or 6.
(f) If last digit of a no. is 9, then last
digit of its complete square root is either 3 or 7.
3.
Prime Numbers
(a) Find the approx square root of given no.
Divide the given no. by the prime no. less than approx square root of no. If a
given no. is not divisible by any of these prime no. then the no. is prime
otherwise not.
(i) To check 359 is a prime number or not.
Approx sq. root = 19
Prime no. < 19 are 2, 3, 5, 7, 11, 13, 17
359 is not divisible by any of these prime
nos. So 359 is a prime no.
(ii) Is 25001+125001+1 is prime or not?
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Divisibility Rules |
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|
Numbers |
IF
a Number |
Examples |
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|
Divisible
by 2 |
End
with 0,2,4,6,8 are divisible by 2 |
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|
Divisible
by 3 |
Sum
of its digits is divisible by 3 |
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|
Divisible
by 4 |
Last
two digit divisible by 4 |
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|
Divisible
by 5 |
Ends
with 0 or 5 |
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|
Divisible
by 6 |
Divides
by Both 2 & 3 |
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|
Divisible
by 7 |
Multiply
the last digit with 2 and subtract it to remaining number in given number,
result must be divisible by 7 |
99995
: 9999 – 2 × 5 = 9989 |
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|
Divisible
by 8 |
Last
3 digit divide by 8 |
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|
Divisible
by 9 |
The
sum of the digit of the number is divisible by nine or I.E the result is a
multiple of nine |
14526:
1+4+5+2+6=18 |
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|
Divisible
by 10 |
End
with 0 |
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|
Divisible
by 11 |
In
a number, if difference of sum of digit at even places and sum of digit at
odd places is either 0 or multiple of |
12342111234211 |
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|
Divisible
by 12 |
The
number must be divisible |
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|
Divisible
by 13 |
Multiply
last digit with 4 and |
8765381387653813 |
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|
Divisible
by 14 |
The
number must be divisible |
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|
Divisible
by 15 |
The
number should be divisible |
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|
Divisible
by 16 |
The
number formed by last |
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|
Divisible
by 17 |
Multiply
last digit with 5 and |
294678:
29467 – 5 × 8 = 29427 |
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|
Divisible
by 18 |
The
number should be divisible |
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|
Divisible
by 19 |
Multiply
last digit with 2 and |
149264:
4 × 2 + 6 = 14 |
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|
Divisible
by 20 |
The
number formed by last |
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