Modulo 100.

Number Theory Level 5

N is number of distinct remainders when squares of numbers from 1 to 99 are divided by 100.Find N .


The answer is 22.

Ramprasad Rakhonde by Ramprasad Rakhonde , 22, India

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1 solution

I found that the last two digits of all numbers from 1 to 50 is same as that from 51-99 and 12 has same remainder as 62 ,,....., 42 has same remainder as 92 and 49 has same remainder as 99

AND THE POSSIBLE LAST TWO DIGITS ARE 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 21 , 69 , 25 , 56 , 89 , 24 , 61 , 84 , 29 , 76 , 44 1, 4,9,16,25,36,49,64,81,21,69,25,56,89,24,61,84,29,76,44

SO THERE ARE 22 22 DIGITS IN TOTAL

HENCE ANSWER IS 22 \boxed{22}

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Another approach is that there are 2 quadratic residues mod 4 and 11 quadratic residues mod 25, hence a total of 22 quadratic residues mod 100 by the Chinese remainder theorem.

Calvin Lin Staff - 6 years, 6 months ago

last two digits of squares of numbers in the form (50-x),(50+x) and (100-x), where x belongs to [1,25], are same as that of square of x

Vighnesh Raut - 6 years, 3 months ago

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