1. Imagine you have 100029 marbles.Your friend has 11(x) marbles, you don't know what x is.
You compare marbles and discover that your friend has "n" less marbles than you. What is "n" if it is less than 12?
2.10 brothers have had an extremely good apple harvest. The 10 brothers have divided up the crop evenly when one brother suddenly has to do an errand. The remaining 9 divide up the crop evenly again when another brother walks off again. This repeats until one brother is left with all the apples. Every time a brother walked off the apples were divided evenly, and every time the brother that walked off did not eat an apple. What is the least amount of apples where this is possible?
9093 for 1.
ReplyDelete2520 for 2.
AB: You got 2 correct, but n is less than 12 for 1.
ReplyDeleteHi Dash! Alex worked on your nice problems. He gets 6 for 1 and also 2520 for 2. -- Geo
ReplyDeleteYou got it right!
ReplyDeleteHere's why:
1. This is the same as asking what 100029 is in mod 11. First, we find out what 100000 is in mod 11. We know that 100 is 1(mod 11)(since 100-1=99, and 99=11*9, or 0(mod 11)) and 10000 is 100*100. This is the same as 1*1, or 1. But we must multiply 10 again, as it is not 10000 but 100000.So it is 1*10 or 10(mod 11).
Now, we know that 29=22+7=(2*11)+7=0+7=7(mod 11). So 100029 is 100000+29=10+7=17=11+6=0+6=6(mod 11).
2.This is the same as asking what is the smallest number divisible by 1 through 10. To find this, you multiply the factors of the numbers together. Why the factors? Then you can see which numbers you already counted and therefore leave them out.
So, first you would write down 2.Then you would multiply 3. Then you notice that 4 is 2*2, so you square the 2. T.hen you multiply 5. 6 is accounted for as 2*3, so you leave it out. You multiply 7. 8 is just 2^3, so you multiply 2 again.You square the 3 to get 9, and 10 is accounted for as 5*2. This comes out to 2^3*3^2*5*7=8*9*5*7=2520.