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adelina 88 [10]
3 years ago
10

Prove that one of every three consecutive positive integer is divisible by 3

Mathematics
1 answer:
Inessa05 [86]3 years ago
5 0
Hello : 
all n in N ; n(n+1)(n+2) = 3a    a in  N  or : <span>≡ 0 (mod 3)
1 ) n </span><span>≡ 0 ( mod 3)...(1)

     n+1 </span>≡ 1 ( mod 3)...(2)
      n+2 ≡ 2 ( mod 3)...(3)
by (1), (2), (3)  : n(n+1)(n+2) ≡ 0×1×2   ( mod 3)   : ≡ 0 (mod 3)
2) n ≡ 1 ( mod 3)...(1)

     n+1 ≡ 2 ( mod 3)...(2)
      n+2 ≡ 3 ( mod 3)...(3)
by (1), (2), (3)  : n(n+1)(n+2) ≡ 1×2 × 3  ( mod 3)   : ≡ 0 (mod 3) , 6≡ 0 (mod)
 3) n  ≡ 2 ( mod 3)...(1)

     n+1 ≡ 3 ( mod 3)...(2)
      n+2 ≡ 4 ( mod 3)...(3)
by (1), (2), (3)  : n(n+1)(n+2) ≡ 2×3 × 4  ( mod 3)   : ≡ 0 (mod 3) , 24≡ 0      (mod3)
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Answer:

The distance between (-9, 8) and (-3, -1) will be:

\:d=3\sqrt{13} units

Step-by-step explanation:

Given the points

  • (-9, 8)
  • (-3, -1)

Finding the distance between (-9, 8) and (-3, -1)

d\:=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}

   =\sqrt{\left(-3-\left(-9\right)\right)^2+\left(-1-8\right)^2}

   =\sqrt{\left(-3+9\right)^2+\left(-1-8\right)^2}

   =\sqrt{36+81}

   =\sqrt{117}

   =3\sqrt{13} units

Therefore, the distance between (-9, 8) and (-3, -1) will be:

\:d=3\sqrt{13} units

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3 years ago
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Naya [18.7K]
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Doreen bought a cell phone for x dollars. One year later the value of the phone was 0.75x. Which expression is another way to de
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3/4x

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The person pays Social Security 6.2 percent of the first $90,000. Medicare is 1.45 percent of all income. Use the tax tables for
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Step-by-step explanation

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2 years ago
During a blizzard it snowed 5 and 8/10in after a week the sun had melted 3 and 1/2 inches of snow how many inches of snow is lef
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2\frac{3}{10} inches of snow is left

<em><u>Solution:</u></em>

Given that,

From given,

Original\ amount\ of\ snow = 5\frac{8}{10}\ inches = \frac{10 \times 5 + 8}{10} = \frac{58}{10}\ inches

Sun\ melted = 3\frac{1}{2}\ inches = \frac{2 \times 3 + 1}{2} = \frac{7}{2}\ inches

<em><u>how many inches of snow is left</u></em>

Snow left = original amount of snow - sun melted

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