Have given below the answers to the world cup questions. The questions can be found here .
1. Australia, in one of their matches, scored a total that is the lowest multiple of 11 with exactly 10 factors. Against who was this?
2^4 * 11 = 176. Australia scored this against Pakistan
2. In one of the matches, the top-scorer (who scored a century) for the team batting first scored 3/8th the team's runs, but the team still did not cross 300. The HCF of the team's score and the top-scorers score is a prime number. Which match was this?
Top-scorer's score is a multiple of 3 greater than 100. So, it has to be a score like 34 * 3, 35 * 3, 36 * 3,..... The overall score is less than 300. So, the overall score has to be lower than 38 * 8. So, the overall score has to be of the form 37 * 8, 36 * 8, 35 * 8,...
The HCF is a prime number, this implies that the top scorer scored 111, the team scored 296. India vs. South Africa. Sachin Tendulkar century
3. A 'minnow' scored a total that can be made a perfect square if it is multiplied by 23. If they had scored 1 more, then their score multiplied by 13 would have yielded a perfect square. What was their score? Bonus point, what match was this?
The score should have been 23 * 4 or 23 * 9, 23 * 9 = 207. 207 + 1 = 208, which is 13 * 16.
India vs. Ireland. Ireland scored 207
4. In a particular match, team A batting first scored twice a perfect square. Team B batting second, also scored twice a perfect square. The total score of both the teams put together was also a perfect square. What match are we talking about here?
India 338, England 338 (169 * 2). Total 676 (26* 26)
5. Team scored a three digit score 'abc' in a match. a and b are prime numbers. a + 2b and a + 2c are also prime. b + 2a and b + 2c are also primes. a + c and b + c are also primes. Two digit numbers 'ab' and 'ba' are both primes. Which match was this?
a + c and b + c are also primes => a, b should be odd, c should be even
a,b cannot be 5 as ab is also prime. So, abh has to be 37.
c can only be zero.
abc = 370 (India vs. Bangladesh)
Labels: CAT number theory, CAT Solutions, World Cup