In a digital encryption protocol, a security key is generated based on a master integer N.… — State PSC Exam

In a digital encryption protocol, a security key is generated based on a master integer $N$ . When $N$ is successively divided by $6$ , $8$ , and $9$ , the resulting remainders are $4$ , $3$ , and $7$ , respectively. If $N$ is the smallest possible positive integer that satisfies these conditions, what is the remainder when $N$ is divided by $19$ ?

Cevap: 16

Cevap

By applying the rules of successive division and working backwards from a final quotient of 0, the smallest possible integer

N

is found to be 358. Dividing 358 by 19 yields a quotient of 18 and a remainder of 16.

Adım Adım Çözüm

Set up equations based on the definition of successive division.

N = 6q_1 + 4

q_1 = 8q_2 + 3

, and

q_2 = 9q_3 + 7

Successive division means each division is performed on the quotient of the previous step.

Determine the value of the final quotient

q_3

to minimize

N

q_3 = 0

The smallest possible positive initial number

N

is obtained when the final successive quotient is zero.

Solve for intermediate quotient

q_2

q_2 = 9(0) + 7 = 7

Substitute

q_3 = 0

into the equation

q_2 = 9q_3 + 7

Solve for intermediate quotient

q_1

q_1 = 8(7) + 3 = 59

Substitute

q_2 = 7

into the equation

q_1 = 8q_2 + 3

Calculate the smallest positive integer

N

N = 6(59) + 4 = 358

Substitute

q_1 = 59

into the first equation.

Divide

N

19

to find the final remainder.

358 = 19 \times 18 + 16

. The remainder is

16

The problem asks for the remainder when the resulting

N

is divided by

19

Anahtar Kavram

Successive Division and Remainder Theorem

Tahmini Süre:1m 30s

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