A student did not notice that the multiplication sign between two 7−digits numbers amd wrote one 14−digits number which turned out to be 3 times the would be product. What are the initial numbers ?


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Question Number 142920 by PRITHWISH SEN 2 last updated on 07/Jun/21

$A student did not notice that the multiplication$ $sign between two 7 - digits numbers amd wrote$ $one 14 - digits number which turned out to be$ $3 times the would be product . What are the initial$ $numbers ?$
Commented by PRITHWISH SEN 2 last updated on 07/Jun/21

$thank you sir$
Commented by mr W last updated on 07/Jun/21

$o n l y o n e s o l u t i o n :$ $1 666 667 \times 3 333 334 = \frac{16 666 673 333 334}{3}$
	Answered by mr W last updated on 07/Jun/21

	$s a y t h e 7 d i g i t n u m b e r s a r e a a n d b .$ $t h e 14 d i g i t n u m b e r i s t h e n$ $\underline{a b} = a \times 10^{7} + b$ $3 \times a \times b = a \times 10^{7} + b$ $b (3 a - 1) = a \times 10^{7}$ $3 a - 1 = \frac{10^{7}}{k} = 3 a - 1 ⩾ 3 \times 1000000 - 1 \Rightarrow k ⩽ 3$ $3 a - 1 = \frac{10^{7}}{k} = 3 a - 1 ⩽ 3 \times 9999999 - 1 \Rightarrow k ⩾ 1$ $w i t h k = 1 :$ $3 a - 1 = \frac{10^{7}}{1} \Rightarrow a \notin N$ $w i t h k = 2 :$ $3 a - 1 = \frac{10^{7}}{2} \Rightarrow a = 1 666 667 ✓$ $\Rightarrow b = k a = 3 333 334$ $o n l y s o l u t i o n i s :$ $1666667 \times 3333334 = 16666673333334 / 3$
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