Question-103209

Question Number 103209 by nimnim last updated on 13/Jul/20

Answered by bramlex last updated on 13/Jul/20

n = 7 k + 1 \dots (1) \times 6

n = 6 p + 3 \dots (2) \times 7

n = 5 q + 2 \dots (3)

\Rightarrow (2) - (1)

\Rightarrow n = 42 (p - k) + 15 \dots (4)

set p - k = l

consider

5 q + 2 = 42 l + 15

\Rightarrow 2 (m o d 5) = 42 l + 15

\Rightarrow 2 l = 2 (m o d 5), l = 1 (m o d 5)

\Leftrightarrow l = 5 t + 1

n = 42 l + 15 = 42 (5 t + 1) + 15

n = 210 t + 57

Commented by nimnim last updated on 13/Jul/20

t h a n k s

Answered by floor(10²Eta[1]) last updated on 13/Jul/20

(1) x \equiv 1 (\mod 7) \Rightarrow x = 7 a + 1

(2) x \equiv 3 (\mod 6)

(3) x \equiv 2 (\mod 5)

(2) : 7 a + 1 \equiv 3 (\mod 6) \Rightarrow a \equiv 2 (\mod 6)

\Rightarrow a = 6 b + 2

\Rightarrow x = 7 (6 b + 2) + 1 = 42 b + 15

(3) : 42 b + 15 \equiv 2 (\mod 5) \Rightarrow 2 b \equiv 2 (\mod 5)

\Rightarrow b \equiv 1 (\mod 5) \Rightarrow b = 5 c + 1

\Rightarrow x = 42 (5 c + 1) + 15

\Rightarrow x = 210 c + 57, c \in Z

Commented by nimnim last updated on 13/Jul/20

T h a n k s

Answered by 1549442205 last updated on 13/Jul/20

From the condition that number when

divided by 6 gives remainder 3 and

when divided by 5 gives remainder 2

it follows that if adding that number

to 3 we get a multiple of 30 . Hence

Denoting by n the number we need find

then n = 30 k - 3 . On the other hands, n = 7 m + 1 . We

infer 30 k + 3 = 7 m + 1 \Rightarrow m = \frac{30 k - 4}{7} = 4 k + \frac{2 k - 4}{7}

Put \frac{2 k - 4}{7} = p \Rightarrow k = \frac{7 p + 4}{2} = 3 p + 2 + \frac{p}{2}

Put \frac{p}{2} = q \Rightarrow p = 2 q \Rightarrow k = 7 q + 2

m = 30 q + 8 \Rightarrow n = 210 q + 57 . Thus, the

number we need find is

n = 210 q + 57 (q \in N)

Commented by nimnim last updated on 13/Jul/20

T h a n k s

Facebook Tweet Pin