Number Play — Hard

Q1. For four natural numbers a, b, c, d, consider the eight expressions of the form a ± b ± c ± d. Which of the following is true? Statement A: All eight expressions always have the same parity. Statement B: All eight expressions always give the same numerical value.

• A. Only Statement A is true
• B. Only Statement B is true
• C. Both A and B are true
• D. Neither A nor B is true

Q2. The sum of five consecutive even natural numbers is 5p, where p is the middle number. If 5p = 90, the largest of the five numbers is

• A. 14
• B. 18
• C. 20
• D. 22

Q3. For six natural numbers a, b, c, d, e, f, all expressions of the form a ± b ± c ± d ± e ± f are formed. Which conclusion follows from the parity argument used in the chapter for four numbers?

• A. All such expressions share the same parity
• B. Half of them are even and half are odd
• C. The parity depends on the order of the numbers
• D. Only the expression with all plus signs has a fixed parity

Q4. A natural number leaves remainder 2 when divided by 3 and remainder 2 when divided by 4. Using the LCM rule from the chapter, the number must be of the form

• A. 6n + 2
• B. 7n + 2
• C. 12n + 2
• D. 24n + 2

Q5. Two even numbers are added. Their sum is a multiple of 4 if and only if

• A. Both numbers are multiples of 4, or neither is a multiple of 4
• B. Exactly one of them is a multiple of 4
• C. Both numbers are odd multiples of 2
• D. Both numbers leave remainder 1 when divided by 4

Q6. Consider the statement: 'If a natural number A is divisible by both 9 and 4, then A is divisible by 36.' By the LCM rule this statement is

• A. Sometimes true
• B. Never true
• C. True only when A is even
• D. Always true

Q7. Read the statement: 'If a number A is divisible by 6 and by 4, then A is divisible by 24.' Using the LCM rule from the chapter, the statement is

• A. Always true
• B. Never true
• C. Sometimes true
• D. True only for prime values of A

Q8. Statement I: The sum of a multiple of 6 and a multiple of 9 is always a multiple of 3. Statement II: The sum of a multiple of 6 and a multiple of 9 is always a multiple of 18. Which is correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Q9. Apply the divisibility-by-11 shortcut from to 328105. Starting from the units digit with a '+' sign, what does the alternating sum tell us?

• A. 5 − 0 + 1 − 8 + 2 − 3 = −3, so 328105 is divisible by 11
• B. 5 − 0 + 1 − 8 + 2 − 3 = −3, so 328105 is NOT divisible by 11
• C. 3 − 2 + 8 − 1 + 0 − 5 = 3, so 328105 is divisible by 11
• D. Sum of all digits = 19, so 328105 is not divisible by 11

Q10. To check whether a number is divisible by 24, the chapter suggests testing divisibility by 3 and 8 separately, not by 4 and 6. The reason this works for 3 and 8 but not for 4 and 6 is

• A. 3 and 8 are coprime so LCM(3, 8) = 24, while 4 and 6 share factor 2 so LCM(4, 6) = 12
• B. 3 and 8 are both prime numbers
• C. 4 and 6 are both even and so cannot test divisibility
• D. Any pair of factors of 24 works equally well

Q11. Find the smallest natural number all of whose digits are even (from {0, 2, 4, 6, 8}) and which is divisible by 9.

• A. 108
• B. 180
• C. 288
• D. 468

Q12. For natural numbers a and b, what is the digital root of the expression 9a + 36b + 13?

• A. 4
• B. 5
• C. 9
• D. It depends on a and b

Q13. Start with a number whose digital root is 5. Add 11 repeatedly. The sequence of digital roots that follows is

• A. 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, …
• B. 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, …
• C. 5, 5, 5, 5, … (it stays the same)
• D. 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, …

Q14. In the cryptarithm PQ × 8 = RS (a 2-digit number times 8 gives a 2-digit number, each letter a distinct digit, leading digits nonzero), the value of PQ is

• A. 11
• B. 12
• C. 13
• D. 14

Q15. In the cryptarithm BYE × 6 = RAY, where BYE and RAY are 3-digit numbers and each letter is a distinct nonzero digit, the value of B must be

• A. 1
• B. 2
• C. 3
• D. Any digit from 1 to 9