Primes and Number Theory

Number Theory Level 2

If the the sum of the first 1234 1234 natural numbers (i.e. 1 + 2 + 3 + 4 + ⋯ + 1234 1+2+3+4+\dots+1234 ) is x x , then what is the largest prime factor of x x ?

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The answer is 617.

Armain Labeeb by Armain Labeeb , 20, Bangladesh

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1 solution

Armain Labeeb
Jul 11, 2016

Let the largest prime factor be k k .

We know,

1 + 2 + 3 + 4 + ⋯ + n = n ( n + 1 ) 2 1+2+3+4+\dots+n=\frac{n(n+1)}{2}

Substitute n = 1234 n=1234 and we have,

1 + 2 + 3 + 4 + ⋯ + 1234 = 1234 ( 1235 ) 2 ⟹ x = 617 × 1235 = 5 × 13 × 19 × 617 ∴ k = 617 \begin{aligned} & & 1+2+3+4+\dots +1234 & =\frac { 1234(1235) }{ 2 } \\ & \Longrightarrow & x & =\, \, 617\, \times \, 1235 \\ & & & =\, \, 5\, \times \, 13\, \times \, 19\, \times \, 617 \\ & \therefore & k & =\,\,\boxed{617} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Note that 617 is prime.

Armain Labeeb - 4 years, 11 months ago

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