Arithmetic (UPSC) Part- VI

Total Questions: 29

11. AB and CD are 2-digit numbers. Multiplying AB with CD results in a 3-digit number DEF. Adding DEF to another 3-digit number GHI results in 975. Further A, B, C, D, E, F, G, H, I are distinct digits. If E = 0, F = 8, then what is A + B + C equal to? [2023-II]

(a) 6
(b) 7
(c) 8
(d) 9

Correct Answer: (a) 6

Solution:

Given that E = 0, F = 8

Probable Value of (D, G): D + G = 9
∴ (D, G) = (0, 9), (1, 8), (2, 7), (3, 6), (4, 5)
Among those value, 0, 6, 7, 8 are assigned to other letters
So, (D, G) = (4, 5)
Now, AB When B = 2, A = 1 and C = 3

∴ A = 1; B = 2, C = 3

Hence, A + B + C = 1 + 2 + 3 = 6

12. Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X? [2024-II]

(a) 2
(b) 4
(c) 6
(d) 8

Correct Answer: (d) 8

Solution:Let number X = 10a + b and Y = 10b + a.
(X + Y) = (10a + b) + (10b + a)
99 = 11 (a + b) {99 is greatest 2-digit number}
(a + b) = 9

Possible (a, b) = (8, 1), (7, 2), (6, 3), (5, 4), (4, 5), (3, 6), (8, 7) and (1, 8)

13. Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements: [2024-II]

(p − r)²(qs) is even
(q − s) q²s is even
(q + r)²(p + s) is odd

Which of the statements given above are correct?

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3

Correct Answer: (d) 1, 2 and 3

Solution:p, q, → Odd number.
r, s → Even number,
Now,
1. (odd — even)² (odd × even)

= (odd)² x (even) = Even
2. (odd — even) x (odd)² x even
= odd x odd x even = even
3. (odd + even)² x (odd + even)
= (odd) x (odd) = odd.
Hence, All three are correct

14. 222³³³ + 333²²² is divisible by which of the following numbers? [2024-II]

(a) 2 and 3 but not 37
(b) 3 and 37 but not 2
(c) 2 and 37 but not 3
(d) 2, 3 and 37

Correct Answer: (b) 3 and 37 but not 2

Solution:(222)³³³ + (333)²²²
{(2 x 111)³} + {(3 x 111)²}³
{2³ x (111)³} + {(3² x 111)³}
{8 x (111)³} + {(9 x (111)³}
(111²)ⁱ[(8 x 111)²⁴; (9y"]
(111P²[(8 x 111) + (9y"]
As (111) is divisible by 3 and 37 but not 2.
.. Given expression is divisible by 3 and 37 but not 2.

15. 421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1? [2024-II]

(a) 1
(b) 2
(c) 3
(d) 4

Correct Answer: (c) 3

Solution:Here, the numbers are 421 and 427. When we subtract
421 and 427 by 1, we get 420 and 426.
Now by taking the prime factors we get: 420 = 2 x 2 x
3 x 5 x 7 x 1. 426 = 2 x 3 x 71 x 1.
The common factors among them are 2, 3 and 6 (product of 2 and 3).
Hence, we can say that 3 numbers can be used as the divisor.

16. Consider the following statements in respect of the sum S = x + y + z, where x, y and z are distinct prime numbers each less than 10: [2024-II]

The unit digit of S can be 0.
The unit digit of S can be 9.
The unit digit of S can be 5.
Which of the statements given above are correct?

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3

Correct Answer: (c) 1 and 3 only

Solution:Prime number (< 10) = 2, 3, 5, 7
S1: s = 2 + 3 + 5 = 10 (correct)
S2: s = 3 + 5 + 7 = 15 (correct)

17. How many consecutive zeros are there at the end of the integer obtained in the product 1² × 2⁴ × 3⁶ × 4⁸ ×...× 25⁵⁰? [2024-II]

(a) 50
(b) 55
(c) 100
(d) 200

Correct Answer: (d) 200

Solution:Given product: 1² x 2⁴ x 3⁶ x 4⁸ x ... (25)⁵⁰
exponent on factor of 10.
(10)²⁰, (20)⁴⁰.
Sum of exponents = 20 + 40 = 60.
exponent on 5 are (5)¹⁰, (15)³⁰, (25)⁵⁰
Sum of exponent = 10 + 30 + 2 x 50 = 140.
∴ Total number of zero = 140 + 60 = 200.

18. A Question is given followed by two Statements I and II. Consider the Question and the Statements. Age of each of P and Q is less than 100 years but more than 10 years. If you interchange the digits of the age of P, the number represents the age of Q. [2024-II]

Question:
What is the difference of the ages?
Statement–I:
The age of P is greater than the age of Q.
Statement–II:
The sum of their ages is $116\frac{11}{6}$ times their difference.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
(b) The Question can be answered by using either Statement alone
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together

Correct Answer: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

Solution:Given that age of P and Q is less than 100 but more than 10.

This implies both P and Q are two-digit numbers. So, 10
< P, Q < 100

Also, let P = xy, then Q = yx

Statement-I alone;

P > Q

There are various possibilities: 81 > 18, 72 > 27... and so on. It alone is not sufficient.

Statement II alone: P + Q = (11/6) (P - Q)

Or 10x + y + 10y + x = (11/6) (10x + y — 10y — x) or
11 (x + y) = (11/6) (9x — 9y) or
6 (x + y) = 9 (x - y) or
2x + 2y = 3x - 3y or
x = 5y

As, x and y must be one-digit numbers, y must be
1 and x there must be 5.

So, P = 51 and Q = 15

So, difference in their ages = P — Q = 51 — 15 = 36 years
So, statement - II alone is sufficient to answer this question.

Hence, the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.

19. Question is given followed by two Statements I and II. Consider the Question and the Statements. [2024-II]

Question:
What are the unique values of x and y, where x, y are distinct natural numbers?

Statement–I: is odd
Statement–II:
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
(b) The Question can be answered by using either Statement alone
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together

Correct Answer: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

Solution:From statement-II,
x.y = 12,
(x, y) = (1, 12), (2, 6), (3, 4), (6, 2), (12, 1).
From statement - I
x
— = odd
y
.. (x, y) = Both are odd number.
or (x, y) = Both are even number.
Thus from both statements, we get that (x, y) = (6, 2)

20. A Question is given followed by two Statements I and II. Consider the Question and the Statements. [2024-II]

Question:
If the average marks in a class are 60, then what is the number of students in the class?

Statement–I:
The highest marks in the class are 70 and the lowest marks are 50.

Statement–II:
Exclusion of highest and lowest marks from the class does not change the average.
Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
(b) The Question can be answered by using either Statement alone
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together

Correct Answer: (d) The Question cannot be answered even by using both the Statements together

Solution:Let number of students = n
∴ Total mark = 60n
From S1 and S2, we have
60 (n — 2) = 60n — 70 — 50
6n — 120 = 6n — 120

Thus, we can not find number of students

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