Free Printable IAT III

1.

When the LDE is solvable

a)

$\text{d|c}$

b)

$\text{c|d}$

c)

$\text{d|a}$

d)

$\text{d|b}$

2.

The LDE has ---- number of solution

a)

1

b)

2

c)

infinite

d)

None of the above

3.

In Fibonacci series which one is the 9^th position

a)

21

b)

25

c)

34

d)

45

4.

a\equiv a\left(mod\ m\right)

is called

a)

Reflexive

b)

Transitive

c)

Symmetric

d)

Anti Symmetric

5.

The product of four consecutive integers is divisible by

a)

5

b)

10

c)

12

d)

14

6.

When the linear congruence

$ax\equiv b\left(mod\ m\right)$ has a unique solution

a)

$\left[a,\ m\right]=1$

b)

$\left(a,\ m\right)=1$

c)

$\left(a,\ m\right)\ne1$

d)

$\left(a,\ m\right)=d$

7.

The congruence

$12x\equiv24\left(mod\ 6\right)$ has how many number of incongruent solution

a)

6

b)

4

c)

2

d)

1

8.

Using casting out nines, find the value of 68464

a)

2

b)

0

c)

5

d)

1

9.

In CRT the moduli are

a)

Prime

b)

Composite

c)

Pairwise prime

d)

Pairwise relatively prime

10.

check the following statement: The linear system has a more number of solution modulo if and only if

\left(\Delta,\ m\right)=1

a)

Statement is correct

b)

Statement is wrong

11.

A palindrome with an even number of digits is

a)

divisible by 10

b)

Not divisible by 11

c)

divisible by 11

d)

not divisible by 10

12.

Which one of the following statement is correct

a)

Every odd integer is congruent to 1 or 3 modulo 4

b)

The square of every integer is congruent to 1 modulo 4

c)

Both a and b

d)

None of the above

13.

What is the remainder of 5! Is divided by 15

a)

0

b)

3

c)

1

d)

5

14.

Whether the LDE

$1076x+2076y=3076$ is solvable

a)

Yes, it is solvable

b)

It is not solvable

c)

Not defined

15.

Whether the number 548152 is divisible by 11

a)

Yes

b)

No

16.

Whether the statement is true or false. If the congruence

$x^2\equiv1\left(mod\ m\right)$ has exactly two solutions, then m is a prime.

a)

True

b)

False

17.

If $p$ is a prime, then $\left(p-1\right)!\equiv-1\left(mod\ p\right)$ is called

a)

Fermat’s theorem

b)

Wilson’s theorem

c)

Euler’s theorem

d)

Multiplicative function

18.

Find the remainder when $24^{16}$ is divided by 17.

a)

1

b)

-1

c)

2

d)

3

19.

Which one is wrong

a)

Let $p$ be a prime and a any positive integer. Then $a^p\equiv a\left(mod\ p\right)$

b)

If p is a prime and p be odd, then $2\left(p-3\right)!\equiv-1\left(mod\ p\right)$

c)

A positive integer $n\ge2$ is a prime if and only if $\left(n-2\right)!\equiv-1\left(mod\ n\right)$

d)

All the above

20.

Is the statement $\left(12+18\right)^{17}\equiv12^{11}+18^{17}$ correct

a)

Yes

b)

No

21.

A number theoretic function f is multiplicative if

a)

f\left(mn\right)=f\left(m\right)f\left(n\right)

b)

$f\left(mn\right)=\frac{f\left(m\right)}{f\left(n\right)}$

c)

$f\left(mn\right)=f\left(m\right)-f\left(n\right)$

d)

$f\left(mn\right)=f\left(n\right)$

22.

Let $\left(a,\ m\right)=1$ and then solution of the linear congruence $ax\equiv b\left(mod\ m\right)$ is

a)

$x\equiv a^{\phi\left(m\right)-1}b\left(mod\ m\right)$

b)

$x\equiv\alpha^{\phi\left(p\right)-1}b\left(mod\ p\right)$

c)

$x\equiv a^{\phi\left(m\right)-1}b\left(mod\ p\right)$

d)

$x\equiv a^{\phi\left(m\right)-1}\left(mod\ m\right)$

23.

Which one is correct from the following statements?

a)

If p be a positive integer then $\phi\left(p\right)=p-1$

b)

If p be a prime number then $\phi\left(p\right)=p-1$

c)

If p be a positive integer then $\phi\left(p\right)=p^e-1$

d)

All the above

24.

What is the canonical decomposition of the number 24

a)

7.3

b)

$2^2.6$

c)

2^3.3

d)

$2^2.3^2$

25.

Let m be a positive integer and a any integer with (a, m) = 1 then $a^{\phi\left(m\right)-1}$ is

a)

an inverse of a modulo m

b)

an inverse of $a^m$ modulo m

c)

called Fermat's little theorem

d)

Euler's Phi function

26.

The sigma function denote the ---- of the positive factors of n

a)

multiple

b)

sum

c)

number

d)

function

27.

The sum of the positive factor of $pq$ is

a)

$1+p+q$

b)

$1-p+q+pq$

c)

$p+q+pq$

d)

None of the above

28.

What is the value of $\sigma\left(p\right)$ , where $p$ is a prime

a)

2

b)

1

c)

$p-1$

d)

$p+1$

29.

The Tau and Sigma functions are

a)

Additive

b)

Multiplicative

c)

both a and b

d)

neither a nor b

30.

The value of $\sum_{\text{d|n}}^{ }\phi\left(d\right)$ for $n=9$

a)

7

b)

9

c)

11

d)

13

IAT III - ANT(MCQ)

Total questions: 30

Worksheet time: 30mins

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