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-

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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