Figure It Out: Patterns & Logic

Questions 1-4

1. The sum of four consecutive numbers is 34. What are these numbers? Let numbers be $x, x+1, x+2, x+3$. $4x + 6 = 34$ $4x = 28 \implies x = 7$. Numbers: 7, 8, 9, 10.

2. Suppose p is the greatest of five consecutive numbers. Describe others. If $p$ is greatest: $p-1, p-2, p-3, p-4$.

3. True/False Statements:

• (i) Sum of two even numbers is a multiple of 3.
  • Sometimes True. (e.g., $2+4=6$ Yes, $2+2=4$ No).
• (ii) If not divisible by 18, then not divisible by 9.
  • False. Example: 9 or 27. Divisible by 9 but not 18.
• (iii) If two numbers are not divisible by 6, sum is not divisible by 6.
  • False. Example: 2 and 4. Sum is 6.
• (iv) Sum of multiple of 6 and multiple of 9 is multiple of 3.
  • Always True. Both 6 and 9 are multiples of 3. Sum of multiples of 3 is a multiple of 3.
• (v) Sum of multiple of 6 and multiple of 3 is multiple of 9.
  • Sometimes True. $6 + 3 = 9$ (Yes). $6 + 6 = 12$ (No).

4. Numbers with remainder 2 (div by 3) and 2 (div by 4).

• Number $N = 3a + 2$ and $N = 4b + 2$.
• $N - 2$ is divisible by 3 and 4.
• LCM(3, 4) = 12.
• $N - 2 = 12k \implies N = 12k + 2$.
• Numbers: 14, 26, 38…

Questions 5-8

5. Pebble Riddle

• Rem 1 mod 3.
• Rem 1 mod 5.
• Rem 0 mod 7.
• From first two: $N$ ends in 1 or 6. $N = 15k + 1$.
• Possible values: 16, 31, 46, 61, 76, 91…
• Check divisibility by 7: 91 is divisible by 7 ($7 \times 13$).
• Answer: 91 pebbles.

6. Sum of three numbers (remainder 2 mod 6).

• Let numbers be $6a+2, 6b+2, 6c+2$.
• Sum = $6(a+b+c) + 6$.
• Sum = $6(a+b+c+1)$.
• This is clearly a multiple of 6.
• Answer: True.

7. Remainders of 4779 and 661.

• $4779 \equiv 5 \pmod 7$.
• $661 \equiv 3 \pmod 7$.
• (i) $4779 + 661 \equiv 5 + 3 = 8 \equiv 1 \pmod 7$.
• (ii) $4779 - 661 \equiv 5 - 3 = 2 \pmod 7$.

8. Remainder 2 (div 3), 3 (div 4), 4 (div 5).

• Notice the pattern: Remainder is always (Divisor - 1).
• $N = \text{LCM}(3, 4, 5)k - 1$.
• $\text{LCM}(3, 4, 5) = 60$.
• Smallest number ($k=1$): $60 - 1 = 59$.