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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Find the least positive integer that is congruent to the given sum, product, or power.

a. ( 3 + 19 + 23 + 52 ) ( mod 6 ) b. ( 2 + 17 + 43 + 117 ) ( mod 4 )

c. ( 14 + 46 + 65 + 92 ) ( mod 11 ) d. ( 9 + 25 + 38 + 92 ) ( mod 7 )

e. ( 7 ) ( 17 ) ( 32 ) ( 62 ) ( mod 5 ) f. ( 6 ) ( 16 ) ( 38 ) ( 118 ) ( mod 9 )

g. ( 4 ) ( 9 ) ( 15 ) ( 59 ) ( mod 7 ) h. ( 5 ) ( 11 ) ( 17 ) ( 65 ) ( mod 7 )

i. 43 15 ( mod 4 ) j. 25 38 ( mod 7 )

k. 62 33 ( mod 5 ) l. 52 26 ( mod 9 )

(a)

To determine

The least positive integer that is congruent to (3+19+23+52)(mod6).

Explanation

Formula used:

Theorem: Substitution Properties:

Suppose ab(modn) and cd(modn). Then, a+cb+d(modn) and acbd(modn).

Explanation:

To find the least positive integer that is congruent to (3+19+23+52)(mod6):

Any number is congruent modulo 6 to its remainder, when divided by 6, and therefore,

33(mod6)

191(mod6)

235(mod6)

524<

(b)

To determine

The least positive integer that is congruent to (2+17+43+117)(mod4).

(c)

To determine

The least positive integer that is congruent to (14+46+65+92)(mod11).

(d)

To determine

The least positive integer that is congruent to (9+25+38+92)(mod7).

(e)

To determine

The least positive integer that is congruent to (7)(17)(32)(62)(mod5).

(f)

To determine

The least positive integer that is congruent to (6)(16)(38)(118)(mod9).

(g)

To determine

The least positive integer that is congruent to (4)(9)(15)(59)(mod7).

(h)

To determine

The least positive integer that is congruent to (5)(11)(17)(65)(mod7).

(i)

To determine

The least positive integer that is congruent to 4315(mod4).

(j)

To determine

The least positive integer that is congruent to 2538(mod7).

(k)

To determine

The least positive integer that is congruent to 6233(mod5).

(l)

To determine

The least positive integer that is congruent to 5226(mod9).

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