Smallest!

Number Theory Level 3

Which is the smallest positive integer that leaves remainders 34 and 23 when divided by 56 and 45 respectively?


The answer is 2498.

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Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Margaret Yu
Jun 2, 2016

Let X be the number,

X = 56a +34 = 45b + 23

56a +34 = 45b + 23

56a +11 = 45b

b= 56 a + 11 45 \frac{56a+ 11}{45}

b= a + 11 a + 11 45 a+ \frac{11a+ 11}{45}

b= a + 11 ( a + 1 ) 45 a+ \frac{11(a+ 1)}{45}

Since 45 does not divide 11, then a+1 must be divisible by 45 in order for b to be an integer. The smallest possible a is 44.

Substituting, X = 2498 \boxed{2498}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

good solution..+1

Ayush G Rai - 5 years ago

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