Class 10-Maths ch-17 Modular Arithmetic

In the set of integers modulo 8, 28 $\otimes_{8} 2 =$ ______________.
(1) 0
(2) 1
(3) 2
(4) 3

If $25 \equiv 4 (m o d p),$ where p is a prime number, then p is ________________.
(1) 3
(2) 5
(3) 7
(4) Either (1) or (3)

Solve for x, if $5 x \equiv 0$ (mod 4).
(1) 0
(2) 3
(3) 2
(4) 4

In the set of integers modulo 12, 38 $\oplus_{12} 28 =____________.$
(1) 6
(2) 5
(3) 4
(4) 3

The largest single - digit number that satisfies $14 x \equiv 4 (m o d 3) i s____________.$
(1) 5
(2) 7
(3) 8
(4) 9

If 8-8-2009 is a Saturday, then 15-8-2010 falls on ___________________.
(1) Saturday
(2) Sunday
(3) Wednesday
(4) Thursday

If 23 $\equiv$ 7 (mod x), then which of the following cannot be the value of x ?
(1) 4
(2) 6
(3) 8
(4) 16

Find the remainder when $13^{15}$ is divisible by 5.
(1) 4
(2) 3
(3) 2
(4) 1

Now the time is 1:30 pm. If I woke up 8 hours ago, then I woke up at ______________.
(1) 4:30 am
(2) 5:30 am
(3) 3:30 am
(4) 6:30 am

10.

If 37 $\equiv$ 18 (mod p), where p is a prime number, then find p.
(1) 3
(2) 7
(3) 19
(4) 18

11.

If 15 $\equiv$ 3 (mod x), then which of the following cannot be the value of x ?
(1) 3
(2) 4
(3) 6
(4) 8

12.

Find the remainder when $5^{18}$ is divided by 19.
(1) 1
(2) 4
(3) 11
(4) 17

13.

The largest two-digit number that satisfies $5 x \equiv 6 (m o d 4) i s___________.$
(1) 96
(2) 97
(3) 98
(4) 99

14.

If you were born on 15-4-1993 which was a Tuesday, then on which day of the week did your birthday fall in 1994?
(1) Tuesday
(2) Wednesday
(3) Thursday
(4) Monday

15.

Find the remainder when $11^{12}$ is divided by 7?
(1) 0
(2) 1
(3) 3
(4) 5

16.

If a $\equiv$ b (mod m) and the remainder obtain when 'a' is divided by m is 2, then find the remainder when 'b' is divided by m.
(1) 2
(2) 1
(3) 0
(4) 3

17.

If $x \equiv y$ (mod 2), then which of the following is / are correct?
(A) x is even and y is odd.
(B) Both x and y are odd.
(C) Both x and y are even.
(1) Only (C)
(2) Only (A)
(3) Both (B) and (C)
(4) Both (A) and (B)

18.

If 1-1-2010 is a Friday, then the fifth Sunday of January, 2011 will fall on ___________.
(1) 26th day
(2) 27th day
(3) 29th day
(4) 30th day

19.

Anand started a work on Sunday at 9:30 am. He finished the work after 87 hours. Then he finished the work on ___________.
(1) Wednesday at 11:30 pm
(2) Thursday at 0:30 am
(3) Wednesday at 0:30 am
(4) Thursday at 11:30 pm

20.

Which of the following are the common solutions of 3x $\equiv$ 0 (mod 6) and 2x $\equiv$ o (mod 4)?
(A) 0
(B) 2
(C) 4
(1) both (A) and (B)
(2) Both (A) and (C)
(3) Both (B) and (C)
(4) All of (A), (B) and (C)

21.

If 15x $\equiv$ 2 (mod 3), then which of the following is a possible value of x?
(1) 3
(2) 315
(3) 0
(4) None of these

22.

Which of the following is a common solution for 6x $\equiv$ 0 (mod 8) and 8x $\equiv$ 0 (mod 10) ?
(1) 0
(2) 4
(3) 6
(4) 8

23.

Find the remainder when $2^{24}$ is divided by 35.
(1) 2
(2) 31
(3) 1
(4) 29

24.

Which of the following is correct?
(1) $5 \oplus_{3} 2 \equiv 3 \otimes_{3} 6 (m o d 4)$
(2) $4 \oplus_{3} 2 \equiv 3 \otimes_{4} 5 (m o d 6)$
(3) $5 \oplus_{5} 3 \equiv 6 \otimes_{8} 9 (m o d 3)$
(4) $5 \oplus_{5} 3 \equiv 6 \otimes_{8} 9 (m o d 4)$

25.

Which of the following is / are correct?
(1) $6 \oplus_{4} 3 \equiv 7 \otimes_{9} 8 (m o d 5)$
(2) $10 \oplus_{5} 4 \equiv 9 \otimes_{11} 9 (m o d 11)$
(3) $14 \oplus_{8} 8 \equiv 15 \otimes_{16} 12 (m o d 4)$
(4) $10 \oplus_{5} 4 \equiv 9 \otimes_{11} 9 (m o d 10)$

26.

If x belongs to the set of residues modulo 10, then the common solution of 5 + x $\equiv$ 0 (mod 3) and 6 + x $\equiv$ 0 (mod 5) is ______________.
(1) 1
(2) 2
(3) 4
(4) 5

27.

By which of the following numbers should $3^{5}$ be divided to obtain a remainder 3?
(1) 7
(2) 11
(3) 5
(4) 3

28.

Find the remainder when $6^{11} - 6$ is divided by 11.
(1) 5
(2) 1
(3) 0
(4) None of these

29.

Find x, if 9x $\equiv$ 2 (mod 7).
(1) 1
(2) 2
(3) 3
(4) 4

30.

Find the remainder when $3^{19}$ is divided by 19.
(1) 3
(2) 15
(3) 16
(4) 19

31.

In the set of integers modulo 9, 15 $\otimes_{9}$ 10 = __________.
(1) 3
(2) 6
(3) 0
(4) 1

32.

If 7x $\equiv$ 1 (mod 5), then which of the following is a possible value of x?
(1) 10
(2) 11
(3) 12
(4) None of these

33.

In order to enter her name in the Guinness Book of world records, Sangeeta started singing on Monday at 10:30 am. If she sings continuously for 36 hours, then she will finish her singing on
(1) Tuesday at 10:30 am
(2) Wednesday at 10:30 am
(3) Tuesday at 10:30 pm
(4) Wednesday at 10:30 pm

34.

In the set of integers modulo 17, 19 $\oplus_{17} 15$ _______________.
(1) 0
(2) 1
(3) 2
(4) 3

35.

Which of the following is a common solution of 3x $\equiv$ 2 (mod 5) and 4x $\equiv$ 0 (mod 6)?
(1) 9
(2) 4
(3) 6
(4) 3

36.

Find the remainder when $3^{215}$ is divided by 43.
(1) 35
(2) 28
(3) 33
(4) 30

37.

Kishore reached his school on Monday at 8:30 am, and then immediately started on a tour to GOA. After $106 \frac{1}{2}$ hours, he reached his house. Then, Kishore reached his home on
(1) Saturday at 7 pm
(2) Friday at 6 pm
(3) Saturday at 6 pm
(4) Friday at 7 pm

38.

If 1-8-2012 is wednesday, then find the day on which we shall celebrate our independence Day in the year 2015.
(1) Saturday
(2) Sunday
(3) Friday
(4) Thursday

39.

Find the remainder when $5^{97}$ is divided by 97.
(1) 5
(2) 97
(3) 92
(4) 100

40.

If a $\equiv$ b (mod m), then which of the following is not always true?
(1) $a^{2} \equiv b^{2} (m o d m)$
(2) $a + m \equiv b + m (m o d 2 m)$
(3) $a m \equiv b m (m o d m^{2})$
(4) $a + m \equiv b - m (m o d 2 m)$

41.

If $x^{3} \equiv x$ (mod 3), then x can be ___________.
(1) 2
(2) 5
(3) 4
(4) All of these

42.

A part of Caley's table for $\otimes_{6}$ is given below. Find the values of x, y and z.
$\begin{matrix} \otimes_{6} & 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 0 & 2 \\ 3 & 0 & 3 & 0 & y & 0 \\ 4 & 0 & 4 & z & 0 & x \end{matrix}$
(1) x = 4, y = 3, z = 2
(2) x = 2, y = 4, z = 3
(3) x = 3, y = 4, z = 2
(4) x = 4, y = 2, z = 3

43.

A part of Caley's table for $\oplus_{7}$ is given below. Find the value of p, q, x and y.
$\begin{matrix} \oplus_{7} & 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & p & x \\ 4 & 5 & Q & y & 1 \end{matrix} \begin{matrix} \end{matrix}$
(1) x = y = 0, p = q = 5
(2) x = y = 1, p = q = 6
(3) x = y = 3, p = q = 4
(4) x = y = 0, p = q = 6

44.

The independence Day of Indian in 2007 was celebrated on a Wednesday, then Children's day in 2008 was celebrated on a _____________.
(1) Friday
(2) Saturday
(3) Sunday
(4) Monday

45.

If 13 $\equiv$ 3 (mod p), then p can be ___________.
(1) 2
(2) 5
(3) 10
(4) All of these

46.

If the 1st January of a certain year, which was not a leap year, was a Thursday, then what day of the week was the 31st December of that year?
(1) Monday
(2) Thursday
(3) Sunday
(4) Saturday

47.

If x+10 $\equiv$ 1 (mod 8), then x can be ___________.
(1) 1
(2) 0
(3) 6
(4) 7

48.

The 2-1-2009 is a Friday. The fourth Sunday of January 2010 falls on the ___________.
(1) 23rd day
(2) 24th day
(3) 25th day
(4) 26th day

49.

Which of the following is / are correct ?
(1) $5 \oplus_{2} 4 \equiv 17 \otimes_{5} 3 (m o d 7)$
(2) $6 \oplus_{4} 7 \equiv 19 \otimes_{9} 3 (m o d 3)$
(3) $9 \oplus_{7} 3 \equiv 8 \otimes_{7} 9 (m o d 9)$
(4) $5 \oplus_{2} 4 \equiv 17 \otimes_{5} 3 (m o d 5)$

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