Creepy Crawlies

Number Theory Level 3

Bugs Buster loves collecting bugs in his tank. He has a collection of millipedes and spiders, and has at most 300 bugs. If he assumes that each millipede has 1000 legs and each spider has 8 legs, then there are 123456 legs in his tank. How many spiders does he have?

The answer is 57.

6 solutions

Joshua Crouch
Sep 2, 2013

The maximum number of bugs he can have is 300.

If he has 123 millipedes, that is 123000 legs.

That leaves 456 legs. 456/8=57

57+123 = 180 This is less than 300.

The answer is 57

Moderator note:

Is 57 the only possible solution?

How did you use the condition that "The maximum number of bugs he can have is 300"?

Maximum number of bugs = 300

Maximum possible number of millipedes = 123 (since 123000 is the maximum possible whole number divisible by 1000), which leaves the remainder(456 legs) to 57 spiders. 123 + 57 = 180 (which is less than 300)

If we reduce the number of millipedes to the next largest number divisible by 1000, i.e. 122,000, then, 122 + 182 = 304 (which violates the 300 bugs max)

Continuing this, we might deduce that as we give up a thousand millipede legs in place for spider legs, we actually increase the number of bugs violating further from the 300 bug max, or for every 1 millipede comes 125 spiders. So this leaves the combination of 123 millipedes and 57 spiders to be the only possible one satisfying the condition of a maximum of 300 bugs.

Francis Parcasio - 7 years, 9 months ago

Good job!

Calvin Lin Staff - 7 years, 9 months ago

Ardilo Indragita Bin Aidir
Sep 7, 2013

For example: a = millipedes, b = spider so a + b <=300 the maximum possible number of milipedes are 123. Consequently the number of spider are 57. so he has 180 bugs. Now assume that the possible number of milipedes is less than 123. Consequently the number of spider is more than or equal to 57 + (1000:8)=182 so he has more than or equal to 304 bugs. because the maximum he has is 300 bugs so the possinle number of spiders he has is 57.

John Palm
Sep 3, 2013

if each millipede has 1000 legs, then he has 123456/1000 millipedes = 123 millipedes. the remainder is 456, so he has 456/8 spiders = 57 spiders

Why must he have 123 millipedes?

Calvin Lin Staff - 7 years, 9 months ago

Jomarie Cabuello
Sep 7, 2013

we got 123456 legs all in all.

The most number of millipedes we have is 123, and the remainder is the least number of spiders.

The remaining 456 is divisible by 8, which means it has 456/8 = 57 spiders and 57+123 = 180.

If we decrease the number of millipedes to 122, we got 1456 legs for spiders. With that we will have 182 spiders. And 182 + 122 = 304 which exceeds the condition 300 bugs.

Sanjeev Rao
Sep 2, 2013

Each millipede has 1,000 legs; 456 is not fully divisible by 1000 so we split them up: $123,456 = 123,000 + 456$

To find out how many millipedes are present we divide it by 1,000, since each millipede has 1,000 legs : $123,000 / 1,000 = 123$

Bugs has 123 millipedes as a result.

Each spider has 8 legs, so we divide 456 by 8, to give : $456 / 8 = 57$

Bugs has 57 spiders as a result.

Vineet Advani
Sep 2, 2013

Now, first we know that the millipedes can take care of the thousands digits so there are 456 extra legs for the spiders to take care of.

So 456/8 = 57

that means that there are 57 spiders in the tank

if each millipede has 1000 legs, then he has 123456/1000 millipedes = 123 millipedes. the remainder is 456, so he has 456/8 spiders = 57 spiders

Mohit Patel - 7 years, 9 months ago

Creepy Crawlies

The answer is 57.

6 solutions

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