Number System Questions and Answers

Answer: B

The value is −2.

Factorise 3024 into primes:

\[ \begin{aligned} 3024 &= 2 \times 1512 = 2^{4} \times 189 \\ 189 &= 3^{3} \times 7 \\ 3024 &= 2^{4} \times 3^{3} \times 7 \end{aligned} \]

Comparing with \(p^{4} \times q^{3} \times r\) gives p = 2, q = 3 and r = 7.

\[ 3p - 5q + r = 6 - 15 + 7 = -2 \]

Swapping p and q by mistake (p = 3, q = 2) gives 9 − 10 + 7 = 6, which is not an option, so the factorisation must be read with the fourth power on 2.

Answer: B

The sum is 300.

The primes between 12 and 49 are 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47, ten numbers in all. Adding them in pairs that are easy to total:

\[ \begin{aligned} (13 + 47) + (17 + 43) &= 60 + 60 \\ (19 + 41) + (23 + 37) &= 60 + 60 \\ 29 + 31 &= 60 \\ \text{total} &= 5 \times 60 = 300 \end{aligned} \]

A common slip is to include 49, which is not prime (\(7 \times 7\)), or to drop a prime from the list; 299, 301 and 302 all differ from the correct total, so check the list of ten primes before adding.

Answer: C

The difference is 6453.

For the greatest number, arrange the digits in descending order: 7510. For the least 4-digit number, put the smallest non-zero digit first and then the rest in ascending order: 1057, because a number cannot begin with 0.

\[ \begin{aligned} 7510 - 1057 &= 6453 \end{aligned} \]

Using 0157 as the least number is not allowed, since it is only a 3-digit number. 6093, 6003 and 6531 come from other wrong choices of the two numbers or from slips in subtraction.

Answer: B

The difference is 66942.

The greatest five-digit number from 8, 2, 7, 0, 6 puts the digits in descending order: 87620. The smallest cannot start with 0, so it starts with the smallest non-zero digit, 2, followed by the rest in ascending order: 20678.

\[ 87620 - 20678 = 66942 \]

Putting 0 first to make 02678 is the usual trap; that is only a four-digit number. The other options come from subtraction slips or a wrong choice of the smallest number.

Answer: A

There are 6 twin-prime pairs from 1 to 50.

Twin primes are pairs of primes that differ by 2. The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The pairs differing by 2 are:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)

That is 6 pairs. 23 has no partner (21 and 25 are composite), nor do 37 and 47. Counting 5 or 4 pairs usually comes from missing (3, 5) or (5, 7), since 5 belongs to two pairs.

Answer: A

The difference is 799200.

In 2853847, the first 8 is in the hundred-thousands place and the second 8 is in the hundreds place:

\[ \begin{aligned} \text{place values} &= 800000 \text{ and } 800 \\ 800000 - 800 &= 799200 \end{aligned} \]

The face value of each 8 is the same, but place value depends on position. 792000 and 799920 come from placing the second 8 in the thousands or tens place, and 7999200 adds a digit.

Answer: D

The numeral not written correctly is VXII.

In Roman numerals a smaller symbol is written before a larger one only to subtract, and only I, X and C may be used that way (IV, IX, XL, XC, CD, CM). V, L and D are never subtracted, so 'VX' is not allowed. Twelve is written XII.

The other three are correct: XXV = 10 + 10 + 5 = 25, LXXII = 50 + 10 + 10 + 2 = 72, and LXIII = 50 + 10 + 3 = 63.

Answer: A

Rational numbers are not closed under division.

A set is closed under an operation if the result of the operation on any two members is again a member. For rational numbers \(a\) and \(b\), \(a + b\), \(a - b\) and \(a \times b\) are always rational.

Division fails in one case: \(a \div 0\) is not defined, so it is not a rational number. Because of that single exception, rational numbers are not closed under division, although the non-zero rationals are. Addition, subtraction and multiplication are all closed, so those statements are true.

Answer: B

The digit that cannot occur is 5.

Check each possible units digit \(d\) of the number and the units digit of \(d^{2}\); we need the square's digit to be \(d + 2\).

\[ \begin{aligned} d = 2 &: 2^{2} = 4 = 2 + 2 \\ d = 4 &: 4^{2} = 16, \text{ digit } 6 = 4 + 2 \\ d = 7 &: 7^{2} = 49, \text{ digit } 9 = 7 + 2 \end{aligned} \]

No other digit works (for example \(3^{2} = 9\), not 5, and \(5^{2} = 25\), not 7). So the units digit of the square can be 4, 6 or 9, but never 5, which is why 4, 6 and 9 are wrong answers.

Answer: B

Only statement (d) is true, so the answer is (d) only.

(d) is true: if p is prime, √p cannot be written as a fraction, because p appears to an odd power in any square, so √2, √3 and √5 are all irrational.

The others fail on fractions and negatives. (a) is false: 1/2 exceeds its square 1/4. (b) is false: −1/2 does not exceed its cube −1/8, since −1/8 is greater. (c) is false: 1/2 exceeds its cube 1/8, and −2 exceeds its cube −8. So every option containing (a), (b) or (c) is wrong.