Are you smarter than a fifth grader?

We have a number ${ a }_{ 1 }{ a }_{ 2 }{ a }_{ 3 }....{ a }_{ 2008 }{ a }_{ 2009 }$ .

So, if we multiply $\overline { { a }_{ 2008 }{ a }_{ 2009 } } x54=(10{ a }_{ 2008 }+{ a }_{ 2009 })x54=..68$ , we obtain that the solution are $({ a }_{ 2008 },{ a }_{ 2009 })=\left\{ (4,2),(9,2) \right\}$ . So thats mean that: $\overline { 42 } X46=..32\\ \overline { 92 } X46=..32$

Let's call that number: a.
We have: $54a \equiv 68[100]$ which means $-46a \equiv 68[100]$ hence $46a \equiv -68 \equiv 32[100]$

It can be done very quickly this way

Given that $n \times 54$ when divided by 100 leaves a remainder 68

$\Rightarrow n \times -46$ leaves a remainder 68

$\Rightarrow n \times 46$ leaves a remainder - 68 i.e -68+100 = 32

I think this problem can be solved without any knowledge on number theory......how?????lets see!!!!!!!First........the last no of the given product is 8...4 can form this no. only after multiplication with a number ending with 2 or 7.......here lies the sums beauty!!!!!! '6'of the no 46 when multiplied with any of 2 or 7 yields the last number as 2....!!!!!!So we are confirmed that the last no. of the required product is 2............!!! now the second-last number of the given product is 6.........now let the second last no.of the given number be "x".....so 2 eqations may be formed
4x+(5.2)=(something)6 or 4x+(5.7)=(something)6 but second equation is not possibe as sum of a even and odd no.always yields an even no. so for first eqation the soln of "x"are 4 or 9 as the last digit of (5.2) is 0 so when multiplied by 46 yields the no.as 3........thats it!!!!!!!!!!!!!!!!!!!!!!

yey i did it...but i think that the last two digits could also have been 9 and 2.it would yield the same result that is last 2 digits after multiplying coul have been 52 as well

Cooool Que bro...... @Krishna Ar it was very easy The product of last two digits with two digits doesnt matters how many digits it is... It is clear that last digit of the no. Is 2 and then after let the second last digit be p P 2 ×54 =4P+10 now as second last digit of product is 6 and greatest posiibility of coming 6 is 26 by regarding conditions thus 4P+10=26 thus P=4 There fore last two digit of 2009 digit number is 2 and 4 By order Now multiplying it By 46 We get the desired digits as 2 and 3 by order Thats it.....

COOL problem bro

@Krishna Ar

Will you please delete the question "Are you smarter than a 5th grader- Part 6· Level 4 (110 points) " from your set since I have completed all the other questions from your set but still haven't been able to make it to the leaderboard .

Thanks for the same .