Get back to business IV

Number Theory Level 1

Express 100 as 2 summands, both positive integers, one of them is divisible by 7 and the other is divisible by 11.

Find the absolute difference of these 2 numbers.

Problem by- Euler ,1770

The answer is 12.

3 solutions

Montgomrey Merlin
Sep 23, 2014

56-44=12 easy ... just use 11 table and calculate it

Archiet Dev
Jun 14, 2014

The two integers are 56 & 44

How is it that the questions in this set are all level 1 but the set itself is level 3?

Tan Li Xuan - 6 years, 11 months ago

I knew it! I tried writing my answer as 56-44, and it just said 'Answer had to be an integer'. I tried writing it as 5644, and it was considered wrong. ;_;

Aisha Nasir - 6 years, 11 months ago

Tough Luck ! ummm

Archiet Dev - 6 years, 11 months ago

We could have 133 + (-33) = 100, since 7 * 19 = 133 and 11 * (-3) = -33. Then a = 133 and b = -33, giving us a - b = 166. There are in fact an infinite number of possible solutions, but only one, (namely 56 - 44 = 12), if it is required that the summands are both positive integers. Perhaps the question should specify this in order to avoid any ambiguity.

Brian Charlesworth - 6 years, 11 months ago

Hey!Thanks & sorry for the inconvenience...

Archiet Dev - 6 years, 11 months ago

You're welcome. I did enter 12 on my second try, so it's all good. :)

Brian Charlesworth - 6 years, 11 months ago

They are both "summands"; the two numbers used to create a sum. If it allowed for negative integers, the word summand would not be used because when you subtract a number that number is called the subtrahend. It is specified in the question if you pay attention to the vocabulary. Also, less specific, the problem states you are dividing 100 into two parts. A "part" cannot be less than zero, otherwise it isn't a 'part", and neither of these "parts" can be larger than the whole.

Nicolas Bryenton - 6 years, 10 months ago

Why are the problems titled "Get back to business"???

Krishna Ramesh - 6 years, 12 months ago

Firstly I am a great admirer of Eminem & one of his songs Business has this sentence .& the other reason is because the problem that I post are easy-so ,y'all just solve & earn point.So,I call it as -Get Back to Business...

Archiet Dev - 6 years, 12 months ago

I did it like this

11 times table 11 22 33 44 55 66 77 88 and the 7 times table 7 14 21 28 35 42 49 56 63

Just by looking at the table now, I know that the required numbers are 56 and 44. How? well I know for certain that

64 + 36 = 100 (a right angled triangle based on Pythagoras theorem)

What happens if we reduce one of the numbers by 20, the other must be increased by 20.

64 + 36 + 20 - 20 =100 (64 - 20) + (36 + 20) = 100

So we have the numbers 44 + 56 = 100

and taking the difference yields = 56 - 44 = 12

Sounds Crazy eh? :D

Awnon Bhowmik - 6 years, 11 months ago

I never heard of "Summmands" and then calling them $a$ and $b$ ... Why not just say "If $a+b=100$ " and go from there.

Marta Reece - 3 years ago

why it can't be 77,33

Sanatan Samkalp - 6 years, 12 months ago

Divide 100 into two parts.... your ans. 77 + 33 =110 not equal to 100

Archiet Dev - 6 years, 12 months ago

Dhananjay Singh
Sep 25, 2014

As a>b, a>50 & b<50, by hit and trial.... a-b=56-44=12

Get back to business IV

The answer is 12.

3 solutions

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