Lesson 9 of 10 · 11 min
Numbers, estimates and remainders
NCERT §7.2.2
Back to the first card: 98⁵. The stall also asks two more: which is bigger, (1.01)¹⁰⁰⁰⁰⁰⁰ or 10000? And what is (1.02)¹⁰ to three decimal places?
The lesson in notes
In short
Large powers: write the base as a round number plus or minus a small one. 98⁵ = (100 − 2)⁵ = 10000000000 − 1000000000 + 40000000 − 800000 + 8000 − 32 = 9039207968.
Our own check: 102⁴ = (100 + 2)⁴ = 100000000 + 8000000 + 240000 + 3200 + 16 = 108243216.
Comparing sizes: (1.01)¹⁰⁰⁰⁰⁰⁰ = (1 + 0.01)¹⁰⁰⁰⁰⁰⁰ = 1 + 1000000 × 0.01 + (other positive terms) = 10001 + (positive terms), so it is larger than 10000.
Estimates: for small x, the first terms of (1 + x)ⁿ give a close value. (1.02)¹⁰ ≈ 1 + 10(0.02) + 45(0.02)² = 1 + 0.2 + 0.018 = 1.218, against the true value 1.21899…; each further term is much smaller than the one before.
Remainders: 6ⁿ − 5n leaves remainder 1 on division by 25. Expanding (1 + 5)ⁿ = 1 + 5n + 5²(nC2) + 5³(nC3) + …, so 6ⁿ − 5n = 1 + 25 × (a whole number).
The same method shows 7ⁿ − 6n − 1 is divisible by 36: (1 + 6)ⁿ = 1 + 6n + 36(nC2 + 6·nC3 + …).
The general idea: to find a power's remainder on division by m, write the base as (multiple of m) ± 1 or ± a small number, and expand.