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From the fact that it is divisible by $25$ but not by $2$, we can deduce that it ends with either $25$ or $75$. Now, consider the multiplications modulo $4$: $1 \times 3 \times 5 \times 7 \times 9 \times \dots \times 99 \equiv 1 \times 3 \times 1 \times 3 \times \dots \times 3 \equiv 1^{25} \times 3^{25} \equiv 1^{25} \times (-1)^{25} \equiv -1 \equiv 75 \not\equiv 25$, so the answer is $\boxed{75}$.

I did not use any website. In fact, I never solved such a problem before. I guess you'll just need to be familiar with prime factors and the modulo operator.

@Krishna Jha
–
That is simply the individual remainders left when $4$ divides the given odd numbers which will obviously be $3$ or $1$ and you can easily find out the number of such terms which yield $1$ or $3$ as remainder.

@Budha Chaitanya
–
Because $4$ divides $100$, and looking at the last two digits of a number is essentially considering de remainder when that number is divided by $100$. As $4$ divides $100$, and we know that the number is one less than a multiple of $4$, then the last two digits of the number are also one less than a multiple of $4$. Hence, it's a nice trick to use here, as $75$ is one less than a multiple of $4$, and $25$ is not.

There is no theorem that says "oh, if you multiply the first 50 odd digits then the last 2 digits are 75." Rather, it takes intuition along with a knowledge of mathematical theorems. In this case, somebody figured "okay, well, there are more than 1 factors of "5" in this, but no factors of 2 so it must end in either 25 or 75. Then, to figure out if it was a 75 or 25 at the end, he counted the numbers of factors of 3, because any number ending in 25 when multiplied by 3 ends with 75, and any number ending in 75 multiplied by 3 ends with 25.

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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TopNewestFrom the fact that it is divisible by $25$ but not by $2$, we can deduce that it ends with either $25$ or $75$. Now, consider the multiplications modulo $4$: $1 \times 3 \times 5 \times 7 \times 9 \times \dots \times 99 \equiv 1 \times 3 \times 1 \times 3 \times \dots \times 3 \equiv 1^{25} \times 3^{25} \equiv 1^{25} \times (-1)^{25} \equiv -1 \equiv 75 \not\equiv 25$, so the answer is $\boxed{75}$.

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Argh, you beat me to it xD Well done.

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how these type of shortcuts can be found? I want to know .if in any website it is plz tell me.

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I did not use any website. In fact, I never solved such a problem before. I guess you'll just need to be familiar with prime factors and the modulo operator.

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refer elemantary number theory by David M Burton

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I think you used Chinese remainder theorem.If not then please tell me the complete theorem.

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He did not use Chinese Remainder Theorem. Which part of his solution are you confused with?

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$4$ divides the given odd numbers which will obviously be $3$ or $1$ and you can easily find out the number of such terms which yield $1$ or $3$ as remainder.

That is simply the individual remainders left whenLog in to reply

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$4$ divides $100$, and looking at the last two digits of a number is essentially considering de remainder when that number is divided by $100$. As $4$ divides $100$, and we know that the number is one less than a multiple of $4$, then the last two digits of the number are also one less than a multiple of $4$. Hence, it's a nice trick to use here, as $75$ is one less than a multiple of $4$, and $25$ is not.

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Its 75... courtesy mathematica ... :-P...Bt how do we solve these sort of problems is still a mystery to me..

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There is no theorem that says "oh, if you multiply the first 50 odd digits then the last 2 digits are 75." Rather, it takes intuition along with a knowledge of mathematical theorems. In this case, somebody figured "okay, well, there are more than 1 factors of "5" in this, but no factors of 2 so it must end in either 25 or 75. Then, to figure out if it was a 75 or 25 at the end, he counted the numbers of factors of 3, because any number ending in 25 when multiplied by 3 ends with 75, and any number ending in 75 multiplied by 3 ends with 25.

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