4-digit Prime Divisibility

Without performing actual division and without using calculators, find out if 999559 1021 \dfrac{999559}{1021} is an integer.

No, it is not an integer. Yes, it is an integer.

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3 solutions

Chris Lewis
Aug 30, 2020

999559 = 1000000 441 = 100 0 2 2 1 2 = ( 1000 + 21 ) ( 1000 21 ) = 1021 × 979 \begin{aligned} 999559&=1000000-441 \\&=1000^2-21^2 \\&=(1000+21)(1000-21) \\&=1021\times 979 \end{aligned}

Chew-Seong Cheong
Aug 30, 2020

1021000 1000 × 1021 999559 = 21441 = 20420 20 × 1021 + 1021 999559 = ( 1000 20 1 ) 1021 999559 1021 = 1000 20 1 = 979 \begin{aligned} \overbrace{1021000}^{1000 \times 1021}-999559 & = 21441 = \overbrace{20420}^{20 \times 1021} + 1021 \\ \implies 999559 & = (1000 - 20 - 1)1021 \\ \implies \frac {999559}{1021} & = 1000-20-1 = 979 \end{aligned}

Yes, 999559 1021 \dfrac {999559}{1021} is an integer .

Lâm Lê
Sep 1, 2020

1021 × 1000 = 1021000 (Trying to get as close as possible to 999559 without getting too accurate) 1021\times1000=1021000\text{ (Trying to get as close as possible to }999559\text{ without getting too accurate)}

1021000 999559 = 21441 1021000-999559=21441

1021 × 20 = 20420 (Trying to get as close as possible to 21441 without getting too accurate) 1021\times20=20420\text{ (Trying to get as close as possible to }21441\text{ without getting too accurate)}

21441 20420 = 1021 21441-20420=1021

20420 mod 1021 0 and 1021 mod 1021 0 21441 mod 1021 0 20420 \text{ mod } 1021\equiv0\text{ and }1021\text{ mod }1021\equiv0\Rightarrow21441\text{ mod }1021\equiv0

1021000 mod 1021 0 999559 mod 1021 0 1021000\text{ mod }1021\equiv0\Rightarrow999559\text{ mod }1021\equiv0

Therefore, the answer is \text{Therefore, the answer is} yes, it is an integer \boxed{\text{yes, it is an integer}}

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