Consecutive difficulty

Algebra Level 3

The number 15 can be written as 7 + 8 or 4 + 5 + 6 or 1 + 2 + 3 + 4 + 5 (the sum of 2 consecutive integers, 3 consecutive integers and 5 consecutive integers).

Find the smallest positive integer that can be expressed as the sum of 3 consecutive integers, 5 consecutive integers and 7 consecutive integers.

Taken from 3rd round of the 2013 South African Junior Maths Olympiad

The answer is 105.

2 solutions

Sunil Pradhan
Jul 19, 2014

sum of numbers = middle number × total numbers

3x = 5y = 7z where x, y, z are middle numbers

3x = 5y then x = 5 and y = 3

and 5y = 7z then y = 7 and z = 5

x: y :: 5 : 3 and y : Z :: 7 : 5

making ratio common x : y : z :: (7×5) : (7×3) : (5×3)

x : y : Z :: 35 : 21 : 15

so number is 35 × 3 = 21 × 5 = 15 × 7 = 105

Awesome solution

Victor Spirou - 6 years, 10 months ago

Victor Spirou
Jul 17, 2014

Lowest common multiple of 3, 5 and 7 as we want the lowest integer for which 3 consecutive/ 5 consecutive/ 7 consecutive numbers. We can use the average to get the answer easily as say we want 3 consecutive numbers to equal 3x. (x-1)+(x)+(x+2)=x+x+x= 3x

Victor, the LCM technique is not applicable as if we find the LCM of 2,3 and 5 we get 30 which is not 15, as is given in the question.

Bhanu Bhandari - 6 years, 11 months ago

The LCM technique would have to work for an odd number of consecutive integers as only and odd number of consecutive integers can get an average of an integer. eg 2+3=5 5/2= 2.5

Victor Spirou - 6 years, 11 months ago

Consecutive difficulty

The answer is 105.

2 solutions

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