12 5 / 11 - 6 17/22 + 1 10/33 solve
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Answer:
We are given that X and Y are two integers; both X and Y are greater than 1, X does not equal Y; and X+Y is less than 100. Foo and Bar are two talented course-18 undergrads. Foo is given the sum, and Bar is given the product of these numbers:
Bar says, "I cannot find the numbers."
Foo says, "I was sure that you could not find them."
Bar says, "Then, I found the numbers."
Foo says, "If you could find them, then I also found them."
What are the numbers?
We'll need a list of the prime numbers under 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97
From Bar's first statement, we can assume that X*Y cannot be prime, since both X and Y are greater than 1. Nor can it be the square of a prime, since X is not equal to Y. If X*Y had exactly two proper divisors, then Bar would know the answer, so this eliminates the product of two distinct primes.
From Foo's first statement, we know that X+Y cannot be the sum of two distinct primes. The prime sums under 100 are:
2+2=4, 2+3=5, 2+5=7, 2+7=9, 2+11=13, 2+13=15, 2+17=19, 2+19=21, 2+23=25, 2+29=31, 2+31=33, 2+37=39, 2+41=43, 2+43=45, 2+47=49, 2+53=55, 2+59=61, 2+61=63, 2+67=69, 2+71=73, 2+73=75, 2+79=81, 2+83=85, 2+89=91, 2+97=99, 3+2=5, 3+3=6, 3+5=8, 3+7=10, 3+11=14, 3+13=16, 3+17=20, 3+19=22, 3+23=26, 3+29=32, 3+31=34, 3+37=40, 3+41=44, 3+43=46, 3+47=50, 3+53=56, 3+59=62, 3+61=64, 3+67=70, 3+71=74, 3+73=76, 3+79=82, 3+83=86, 3+89=92, 5+2=7, 5+3=8, 5+5=10, 5+7=12, 5+11=16, 5+13=18, 5+17=22, 5+19=24, 5+23=28, 5+29=34, 5+31=36, 5+37=42, 5+41=46, 5+43=48, 5+47=52, 5+53=58, 5+59=64, 5+61=66, 5+67=72, 5+71=76, 5+73=78, 5+79=84, 5+83=88, 5+89=94, 7+2=9, 7+3=10, 7+5=12, 7+7=14, 7+11=18, 7+13=20, 7+17=24, 7+19=26, 7+23=30, 7+29=36, 7+31=38, 7+37=44, 7+41=48, 7+43=50, 7+47=54, 7+53=60, 7+59=66, 7+61=68, 7+67=74, 7+71=78, 7+73=80, 7+79=86, 7+83=90, 7+89=96, 11+2=13, 11+3=14, 11+5=16, 11+7=18, 11+11=22, 11+13=24, 11+17=28, 11+19=30, 11+23=34, 11+29=40, 11+31=42, 11+37=48, 11+41=52, 11+43=54, 11+47=58, 11+53=64, 11+59=70, 11+61=72, 11+67=78, 11+71=82, 11+73=84, 11+79=90, 11+83=94, 13+2=15, 13+3=16, 13+5=18, 13+7=20, 13+11=24, 13+13=26, 13+17=30, 13+19=32, 13+23=36, 13+29=42, 13+31=44, 13+37=50, 13+41=54, 13+43=56, 13+47=60, 13+53=66, 13+59=72, 13+61=74, 13+67=80, 13+71=84, 13+73=86, 13+79=92, 13+83=96, 17+2=19, 17+3=20, 17+5=22, 17+7=24, 17+11=28, 17+13=30, 17+17=34, 17+19=36, 17+23=40, 17+29=46, 17+31=48, 17+37=54, 17+41=58, 17+43=60, 17+47=64, 17+53=70, 17+59=76, 17+61=78, 17+67=84, 17+71=88, 17+73=90, 17+79=96, 19+2=21, 19+3=22, 19+5=24, 19+7=26, 19+11=30, 19+13=32, 19+17=36, 19+19=38, 19+23=42, 19+29=48, 19+31=50, 19+37=56, 19+41=60, 19+43=62, 19+47=66, 19+53=72, 19+59=78, 19+61=80, 19+67=86, 19+71=90, 19+73=92, 19+79=98, 23+2=25, 23+3=26, 23+5=28, 23+7=30, 23+11=34, 23+13=36, 23+17=40, 23+19=42, 23+23=46, 23+29=52, 23+31=54, 23+37=60, 23+41=64, 23+43=66, 23+47=70, 23+53=76, 23+59=82, 23+61=84, 23+67=90, 23+71=94, 23+73=96, 29+2=31, 29+3=32, 29+5=34, 29+7=36, 29+11=40, 29+13=42, 29+17=46, 29+19=48, 29+23=52, 29+29=58, 29+31=60, 29+37=66, 29+41=70, 29+43=72, 29+47=76, 29+53=82, 29+59=88, 29+61=90, 29+67=96, 31+2=33, 31+3=34, 31+5=36, 31+7=38, 31+11=42, 31+13=44, 31+17=48, 31+19=50, 31+23=54, 31+29=60, 31+31=62, 31+37=68, 31+41=72, 31+43=74, 31+47=78, 31+53=84, 31+59=90, 31+61=92, 31+67=98, 37+2=39, 37+3=40, 37+5=42, 37+7=44, 37+11=48, 37+13=50, 37+17=54, 37+19=56, 37+23=60, 37+29=66, 37+31=68, 37+37=74, 37+41=78, 37+43=80, 37+47=84, 37+53=90, 37+59=96, 37+61=98, 41+2=43, 41+3=44, 41+5=46, 41+7=48, 41+11=52, 41+13=54, 41+17=58, 41+19=60, 41+23=64, 41+29=70, 41+31=72, 41+37=78, 41+41=82, 41+43=84, 41+47=88, 41+53=94, 43+2=45, 43+3=46, 43+5=48, 43+7=50, 43+11=54, 43+13=56, 43+17=60, 43+19=62, 43+23=66, 43+29=72, 43+31=74, 43+37=80, 43+41=84, 43+43=86, 43+47=90, 43+53=96, 47+2=49, 47+3=50, 47+5=52, 47+7=54, 47+11=58, 47+13=60, 47+17=64, 47+19=66, 47+23=70, 47+29=76, 47+31=78, 47+37=84, 47+41=88, 47+43=90, 47+47=94, 53+2=55, 53+3=56, 53+5=58, 53+7=60, 53+11=64, 53+13=66, 53+17=70, 53+19=72, 53+23=76, 53+29=82, 53+31=84, 53+37=90, 53+41=94, 53+43=96, 59+2=61, 59+3=62, 59+5=64, 59+7=66, 59+11=70, 59+13=72, 59+17=76, 59+19=78, 59+23=82, 59+29=88, 59+31=90, 59+37=96, 61+2=63, 61+3=64, 61+5=66, 61+7=68, 61+11=72, 61+13=74, 61+17=78, 61+19=80, 61+23=84, 61+29=90, 61+31=92, 61+37=98, 67+2=69, 67+3=70, 67+5=72, 67+7=74, 67+11=78, 67+13=80, 67+17=84, 67+19=86, 67+23=90, 67+29=96, 67+31=98, 71+2=73, 71+3=74, 71+5=76, 71+7=78, 71+11=82, 71+13=84, 71+17=88, 71+19=90, 71+23=94, 73+2=75, 73+3=76, 73+5=78, 73+7=80, 73+11=84, 73+13=86, 73+17=90, 73+19=92, 73+23=96, 79+2=81, 79+3=82, 79+5=84, 79+7=86, 79+11=90, 79+13=92, 79+17=96, 79+19=98, 83+2=85, 83+3=86, 83+5=88, 83+7=90, 83+11=94, 83+13=96, 89+2=91, 89+3=92, 89+5=94, 89+7=96, and 97+2=99