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PLEASE HELP!!

Hello everybody!! I had my FTRE(FIITJEE TALENT REWARD EXAM) yesterday.This exam is held all over India and I think you all must have understood the purpose!There was this question that I have no idea how to solve and I was thinking if someone could help me.Anyway,the question is,\(\overline{ab}\times \overline{c}=\overline{def}\).These conditions apply on the digits,\((a,b,c,d,e,f):\\ \bullet they\ are\ distinct\\ \bullet none\ of\ them\ is\ equal\ to\ 0\\ \bullet they\ are\ all\ less\ than\ 7.\)Find the digits.

Note by Adarsh Kumar
2 years, 7 months ago

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\[54\times 3=162\] Aneesh Kundu · 2 years, 7 months ago

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@Aneesh Kundu awesome!Could you please tell how you got that. Adarsh Kumar · 2 years, 7 months ago

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@Adarsh Kumar

  • Firstly since all the digits are distinct so \(b, c, f\not = 5\), moreover \(d_{max}=3\) ( max occurs at \(54\times 6=324\) )so only \(a=5\) or \(e=5\) .

  • \(b,c\not = 1\)(as that would result in repetition)

  • \(f\not = 1\) (as 1 would occur only at \(u(1\times 1)=1\), \(u(7\times 3) =1\), \(u(9\times 9)=1\) none of which satisfy the conditions.)

We start with \(a=5\),

  • If \(b=2\) , then \(c\) can be \(3, 6\)

  • If \(b=3\) , then \(c\) can be \(2, 4\)

  • If \(b=4\) , then \(c\) can be \(3\)

  • If \(b=6\) , then \(c\) can be \(2\)

A quick check gives us the solution.

NOTE

  • In the last step we cross out some values which give us a bigger unit's digit than 6 or if a repetition occurs.

  • \(u(x)\) means unit's digit of \(x\).

Aneesh Kundu · 2 years, 7 months ago

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@Aneesh Kundu Thanx dude! Adarsh Kumar · 2 years, 7 months ago

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54x3=162 Manit Kapoor · 2 years, 6 months ago

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@Manit Kapoor thanku! Adarsh Kumar · 2 years, 6 months ago

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See such questions could by setting few conditions by ourselves-

lets take it that way -- if b =x then c must be what such that f is not equal to c or b

If we put b as 1 then all values of f would be equal to c.

If we put b as 2 we see that c could be = 3

If we put b as 3 we see that c could be = 2 , 4

If we put b as 4 wee see that c could be = 3

5 can never stand on our suppositions

even 6 won't be

so we examine the 4 values of b = 2, 3 , 4 and c also 2 , 3 , 4

then we think for value of a

If a = 1 then it won't be 3 digit number . Similarly even 2 won' t be - Then further we could do trial & error wid the remaining values

hence , 54*3 = 162 Palash Som · 2 years, 7 months ago

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54 *3 = 162 Palash Som · 2 years, 7 months ago

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@Palash Som yup!that is correct! Adarsh Kumar · 2 years, 7 months ago

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