Pairs of students wrote social media posts in which they invited people to the "Math nights". A pair of students which gets the most likes together will be rewarded. Anna got 3741 and Marco got 2754, but claim to have 6705 together, which makes other students suspicious of fraud. If we allow the possibility that the numbers are not written in the decimal number system, can Anna and Marco be correct? Explain
Answers
Given : 3731 + 2754 = 6705
There is possibility that the numbers are not written in the decimal number system
To Find : is the sum possible
Solution:
(3731)₁₀ + (2754)₁₀ = (5485)₁₀ ≠ (6705)₁₀
As 5485 < 6705
Hence sum to be correct base must be less than 10
as digits used are upto 7
Hence possible bases can be 8 or 9
Using base 9
(3731)₉ = 3*9³ + 7*9² + 3*9¹ + 1 * 9⁰ = (2782)₁₀
(2754)₉ = 2*9³ + 7*9² + 5*9¹ + 4 * 9⁰ = (2074)₁₀
(6705)₉ = 6*9³ + 7*9² + 0*9¹ + 5* 9⁰ = (4946)₁₀
2782 + 2074 = 4856 ≠ 4946
Using base 9
(3731)₈ = 3*8³ + 7*8² + 3*8¹ + 1 * 8⁰ = (2009)₁₀
(2754)₈ = 2*8³ + 7*8² + 5*8¹ + 4 * 8⁰ = (1516)₁₀
(6705)₈ = 6*8³ + 7*8² + 0*8¹ + 5* 8⁰ = (3525)₁₀
2009 + 1516 = 3525 = 3525
Hence they are correct if base 8 is used.
as in base 8
(3731)₈ + (2754)₈ = (6705)₈
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