Number Play — Practice

Question 1

Is 13,572 divisible by 6?

Accepted Answer

✦ Answer: Yes. Test: (a) Last digit 2 → divisible by 2. (b) Sum of digits 1+3+5+7+2 = 18; 18 ÷ 3 = 6 → divisible by 3. Since divisible by both 2 and 3, divisible by 6.

Question 2

What is the 10th Fibonacci number?

Accepted Answer

✦ Answer: F₁=1, F₂=1, F₃=2, F₄=3, F₅=5, F₆=8, F₇=13, F₈=21, F₉=34, F₁₀=55. So F₁₀ = 55.

Question 3

Using Vedic method, compute 85².

Accepted Answer

Step 1 — Identify the pattern for numbers ending in 5.
  For ab5²: write a × (a+1) followed by 25.

Step 2 — Apply for 85².
  a = 8
  a × (a+1) = 8 × 9 = 72
  Append 25: 7225

Step 3 — Verify.
  85 × 85: 85 × 85 = 85 × 80 + 85 × 5 = 6800 + 425 = 7225 ✓

✦ Answer: 85² = 7225.

Question 4

Find the magic constant for a 5×5 magic square containing numbers 1 to 25.

Accepted Answer

Step 1 — Apply magic constant formula.
  M = n(n² + 1)/2
  where n = order of the square = 5

Step 2 — Compute.
  M = 5(5² + 1)/2
     = 5(25 + 1)/2
     = 5 × 26 / 2
     = 130 / 2
     = 65

Step 3 — Verify intuition.
  Sum of 1 to 25 = 25 × 26 / 2 = 325
  Each row should sum to 325/5 = 65 ✓

✦ Answer: Magic constant for 5×5 = 65. Every row, column, diagonal sums to 65.

Question 5

(a) Find the sum 1 + 2 + 3 + ... + 200. (b) Find the sum of squares 1² + 2² + 3² + ... + 20². (c) Find the sum of cubes 1³ + 2³ + 3³ + ... + 10³.

Accepted Answer

Part (a) — Sum 1 to 200.
  Formula: n(n+1)/2
  = 200 × 201 / 2
  = 100 × 201
  = 20,100

Part (b) — Sum of squares 1² to 20².
  Formula: n(n+1)(2n+1)/6
  = 20 × 21 × 41 / 6
  = 17,220 / 6
  Wait, let me recompute: 20 × 21 = 420; 420 × 41 = 17,220
  17,220 / 6 = 2870

Part (c) — Sum of cubes 1³ to 10³.
  Formula: [n(n+1)/2]²
  = [10 × 11 / 2]²
  = 55²
  = 3,025

✦ Answer: (a) 20,100. (b) 2,870. (c) 3,025.

Practice — Number Play

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