Is the sum of the integers from 54 to 153 inclusive, divisib

Question

Is the sum of the integers from 54 to 153 inclusive, divisible by 100?

A. Yes
B. No.

Hint: Can be solved fast with a property.

Bunuel · Accepted Answer

Correct. Properties of consecutive integers: • If n is odd, the sum of n consecutive integers is always divisible by n. Given \{9,10,11\} , we have n=3=odd consecutive integers. The sum is 9+10+11=30, which is divisible by 3. • If n is even, the sum of n consecutive integers is never divisible by n. Given \{9,10,11,12\} , we have n=4=even consecutive integers. The sum is 9+10+11+12=42, which is NOT divisible by 4. For more check here: math-number-theory-88376.html

jlgdr · Answer

Also by Property:

For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items.

Bunuel · Answer

# of integers from 54 to 153 inclusive is 153-54+1=100.

The sum = (average)*(# of integers) = (54+153)/2*100=103.5*100=10350 --> not a multiple of 100.

pqhai · Answer

My other approach

Sum of 1 to 153, inclusive = x 153 = 77 x 153 Sum of 1 to 53, inclusive = x 53 = 27 x 153 Sum of 54 to 153, inclusive = 77*153 - 27*53 = 77*100 + 77*53 - 27*53 = 77*100 + 53*(77-27) = 77*100 + 53*50 Only 77*100 is divisible by 100 ==> The ans is: NOT divisible by 100 Hope it helps.

stonecold · Answer

Great Question.
I actually calculated the sum without looking for the property.

Also to add to the properties =>
Mean of n consecutives can be of the form  x  or  x.5  (for any integer x)

Nunuboy1994 · Answer

jlgdr  Nice question

Last- First/ rate of increase +1 = # items
 153-54/ 1 +1 - 
 99/1 +1 = 100

54 +153 = 207

According to the property 207 cannot be a multiple of 100; but here's another example 8 9 10 11- there are 4 terms and the sum is 38...38 cannot be a multiple of 4.

Kezia9 · Answer

I solved it using Arithmetic Progression.

bumpbot · Answer

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Is the sum of the integers from 54 to 153 inclusive, divisib

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Is the sum of the integers from 54 to 153 inclusive, divisib

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