Vaishali came across some interesting numbers. She describes them as the numbers of the type 'n' such that nx²y³ where x and y are natural numbers. If she adds all such numbers less than 3000, then which of the following number will she get?
53955
4501500
11025
5794
no fake answers, it will be reported
Answers
Given : Vaishali came across some interesting numbers.
She describes them as the numbers of the type 'n' such that n
x²y³ where x and y are natural numbers.
If she adds all such numbers less than 3000,
To Find : Sum
Solution:
x²y³ < 3000,
x² < 3000
Hence x can vary from 1 to 54
y can vary from 1 to 14
y = 1 => x from 1 to 54
∑n² = n(n + 1)(2n + 1)/6
Sum = 1 ( 54 (54 + 1) ( 2 * 54 + 1) / 6) = 53955
y = 2 => x from 1 to 19
Sum = 2³ ( 19 (19 + 1) ( 2 * 19 + 1) / 6) = 19760
Similarly find till 14³
3³( 1² + 2² + _____ + 10²) = 10395
4³( 1² + 2² + _____ + 6²) = 5824
5³( 1² + 2² + 3² +4²) = 3750
6³( 1² + 2² + 3²) = 3024
7³( 1² + 2² ) = 1715
8³( 1² + 2² ) = 2560
9³( 1² + 2² ) = 3645
10³( 1² ) = 1000
11³( 1² ) = 1331
12³( 1²) = 1728
13³( 1² ) = 2197
14³( 1²) = 2744
Sum of all these numbers = 113628
But these numbers have repeated numbers
64 = 1³ * 8² = 4³ * 1²
256 = 1³ * 16² = 4³ * 2²
576 = 1³ *24² = 4³ * 3²
729 = 1³ *27² = 9³ * 1²
1024 = 1³ *32² = 4³ * 4²
1600 = 1³ *40² = 4³ * 5²
2304 = 1³ *48² = 4³ * 6²
2916 = 1³ *54² = 9³ * 2²
512 = 2³ *8² = 8³ * 1²
2048 = 2³ * 16² = 8³ * 2²
1728 = 3³ * 8² = 12³ * 1²
Adds upto = 13757
Hence total = 113628 - 13757 = 99871
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