Find all the subgroup of ( Z 12 , + ) \left(Z_{12},+\right) ( Z 1 2 , + )

{ 0 , 6 } , { 0 , 4 , 8 } , { 0 , 2 , 4 , 6 , 8 , 10 } \left\{0,\ 6\right\},\left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\} { 0 , 6 } , { 0 , 4 , 8 } , { 0 , 2 , 4 , 6 , 8 , 1 0 }

W h e n t h e f u n c t i o n f i s When\ the\ function\ f\ is\ W h e n t h e f u n c t i o n f i s called\ an\ isomorphism c a l l e d a n i s o m o r p h i s m

A hom ⁡ o g e n e o u s f u n c t i o n i s s t a t i f i e s A\ \hom ogeneous\ function\ is\ statifies\ A hom o g e n e o u s f u n c t i o n i s s t a t i f i e s b o t h o n e − t o − o n e a n d o n t o both\ one-to-one\ and\ onto b o t h o n e − t o − o n e a n d o n t o

f u n c t i o n i s s t a t i f i e s b o t h o n e − t o − o n e a n d o n t o function\ is\ statifies\ both\ one-to-one\ and\ onto f u n c t i o n i s s t a t i f i e s b o t h o n e − t o − o n e a n d o n t o

N o n e o f t h e s e None\ of\ these N o n e o f t h e s e

W h e n t h e r i n g R i s c a l l e d c o m m u t a t i v e r i n g When\ the\ ring\ R\ is\ called\ commutative\ ring W h e n t h e r i n g R i s c a l l e d c o m m u t a t i v e r i n g

I f a b = b a f o r a l l a , b ∈ R If\ ab=ba\ for\ all\ a,b\in R I f a b = b a f o r a l l a , b ∈ R

I f a b ≠ b a f o r a l l a , b ∈ R If\ ab\ne ba\ for\ all\ a,b\in R I f a b  = b a f o r a l l a , b ∈ R

I f a b = b a f o r a l l a , b ∉ R If\ ab=ba\ for\ all\ a,b\notin R I f a b = b a f o r a l l a , b ∈ / R

W h i c h o n e i s t h e c o r r e c t f o r m o f Which\ one\ is\ the\ correct\ form\ of W h i c h o n e i s t h e c o r r e c t f o r m o f D i v i s i o n a lg ⁡ o r i t h m Division\ a\lg orithm D i v i s i o n a l g o r i t h m

f ( x ) = q ( x ) + r ( x ) f\left(x\right)=q\left(x\right)+r\left(x\right) f ( x ) = q ( x ) + r ( x )

How many polynomials are there of degree n n n in Z 11 [ x ] Z_{11}\left[x\right] Z 1 1 [ x ] ?

10 ( 11 ) n 10\left(11\right)^n 1 0 ( 1 1 ) n

Every non empty set of positive integers has a least number.

The inclusion-Exclusion principle

When the linear congruence a x ≡ b ( m o d m ) ax\equiv b\left(mod\ m\right) a x ≡ b ( m o d m ) has a unique solution

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

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

( a , m ) ≠ 1 \left(a,\ m\right)\ne1 ( a , m )  = 1

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

The solution of sun Tsu's puzzle by iteration is

x = 52 + 105 t x=52+105t x = 5 2 + 1 0 5 t

x = 52 − 105 t x=52-105t x = 5 2 − 1 0 5 t

x = − 52 − 105 t x=-52-105t x = − 5 2 − 1 0 5 t

Which one of the following statement is correct

Every odd integer is congruent to 1 or 3 modulo 4

The square of every integer is congruent to 1 modulo 4

A number theoretic function f is multiplicative if

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

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

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

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

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

x ≡ a ϕ ( m ) − 1 b ( m o d m ) x\equiv a^{\phi\left(m\right)-1}b\left(mod\ m\right) x ≡ a ϕ ( m ) − 1 b ( m o d m )

x ≡ α ϕ ( p ) − 1 b ( m o d p ) x\equiv\alpha^{\phi\left(p\right)-1}b\left(mod\ p\right) x ≡ α ϕ ( p ) − 1 b ( m o d p )

x ≡ a ϕ ( m ) − 1 b ( m o d p ) x\equiv a^{\phi\left(m\right)-1}b\left(mod\ p\right) x ≡ a ϕ ( m ) − 1 b ( m o d p )

x ≡ a ϕ ( m ) − 1 ( m o d m ) x\equiv a^{\phi\left(m\right)-1}\left(mod\ m\right) x ≡ a ϕ ( m ) − 1 ( m o d m )

MA8551 / ANT - MODEL II

Authored by Shanmugavalli R

Mathematics

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45 questions

Show all answers

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Find all the subgroup of
$\left(Z_{12},+\right)$

$\left\{0,\ 6\right\},\left\{0,\ 3,\ 6,\ 9\right\},\ \left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

$\left\{0,\ 3,\ 6,\ 9\right\},\ \left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

$\left\{0,\ 6\right\},\left\{0,\ 4,\ 8\right\},\left\{0,\ 2,\ 4,\ 6,\ 8,\ 10\right\}$

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

$When\ the\ function\ f\ is\$ $called\ an\ isomorphism$

$A\ \hom ogeneous\ function\ is\ statifies\$ $both\ one-to-one\ and\ onto$

$function\ is\ statifies\ both\ one-to-one\ and\ onto$

$None\ of\ these$

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

$Deter\min e\ U_{14},\ the\ group\ of\ units\ of\ the\ ring\ \left(Z_n,\ +,\ \circ\right)$

$\left\{1,3,5,7,9,11,13\right\}$

$\left\{1,3,5,9,11,13\right\}$

$\left\{1,3,7,9,11,13\right\}$

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

$When\ the\ ring\ R\ is\ called\ commutative\ ring$

$If\ ab=ba\ for\ all\ a,b\in R$

$If\ ab\ne ba\ for\ all\ a,b\in R$

$If\ ab=ba\ for\ all\ a,b\notin R$

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

$Find\ the\ value\ of\ additive$ $\ identity\ from\ the\ given\ equation$ $x+y=x+y-7,\ x\circ y=x+y-3xy$

$Z=0$

$Z=-1$

$Z=7$

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

$Find\ \left[100\right]^{-1}\ in\ the\ ring\ Z_{1009}$

110

101

111

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

$FInd\ the\ multiplicative\ inverse\ of\ 4\ in\ Z_{11}$

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

$What\ is\ the\ last\ digit\ in\ 3^{55}$

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

$Howmany\ units\ are\ there\ in\ the\ ring\ Z_8?$

10.

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

$Which\ one\ is\ the\ correct\ form\ of$ $Division\ a\lg orithm$

$f\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

$f\left(x\right)=q\left(x\right)g\left(x\right)-r\left(x\right)$

$f\left(x\right)=q\left(x\right)+r\left(x\right)$

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MA8551 / ANT - MODEL II

Find all the subgroup of (Z12,+)\left(Z_{12},+\right)(Z12​,+)

When the function f is When\ the\ function\ f\ is\ When the function f is called\ an\ isomorphismcalled an isomorphism

Determin⁡e U14, the group of units of the ring (Zn, +, ∘)Deter\min e\ U_{14},\ the\ group\ of\ units\ of\ the\ ring\ \left(Z_n,\ +,\ \circ\right)Determine U14​, the group of units of the ring (Zn​, +, ∘)

When the ring R is called commutative ringWhen\ the\ ring\ R\ is\ called\ commutative\ ringWhen the ring R is called commutative ring

Find the value of additiveFind\ the\ value\ of\ additiveFind the value of additive identity from the given equation\ identity\ from\ the\ given\ equation identity from the given equation x+y=x+y-7,\ x\circ y=x+y-3xyx+y=x+y−7, x∘y=x+y−3xy

Find [100]−1 in the ring Z1009Find\ \left[100\right]^{-1}\ in\ the\ ring\ Z_{1009}Find [100]−1 in the ring Z1009​

FInd the multiplicative inverse of 4 in Z11FInd\ the\ multiplicative\ inverse\ of\ 4\ in\ Z_{11}FInd the multiplicative inverse of 4 in Z11​

What is the last digit in 355What\ is\ the\ last\ digit\ in\ 3^{55}What is the last digit in 355

Howmany units are there in the ring Z8?Howmany\ units\ are\ there\ in\ the\ ring\ Z_8?Howmany units are there in the ring Z8​?

Which one is the correct form ofWhich\ one\ is\ the\ correct\ form\ ofWhich one is the correct form of Division alg⁡orithmDivision\ a\lg orithmDivision algorithm

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Discover more resources for Mathematics

Find all the subgroup of
$\left(Z_{12},+\right)$

$When\ the\ function\ f\ is\$ $called\ an\ isomorphism$

$Deter\min e\ U_{14},\ the\ group\ of\ units\ of\ the\ ring\ \left(Z_n,\ +,\ \circ\right)$

$When\ the\ ring\ R\ is\ called\ commutative\ ring$

$Find\ the\ value\ of\ additive$ $\ identity\ from\ the\ given\ equation$ $x+y=x+y-7,\ x\circ y=x+y-3xy$

$Find\ \left[100\right]^{-1}\ in\ the\ ring\ Z_{1009}$

$FInd\ the\ multiplicative\ inverse\ of\ 4\ in\ Z_{11}$

$What\ is\ the\ last\ digit\ in\ 3^{55}$

$Howmany\ units\ are\ there\ in\ the\ ring\ Z_8?$

$Which\ one\ is\ the\ correct\ form\ of$ $Division\ a\lg orithm$