LEARN · EXPLAIN · REVISE

Find the remainder when (1+10)^m is divided by 100.

Read the idea, work independently, then explain what changed.

高三選擇性必修第三册(A版).pdf · 6.3 · PDF 34 / printed page 29

TOPIC 01

Binomial theorem

Use the general term to locate coefficients, constants and coefficient sums.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

m=8

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Decide whether choices are alternative cases or successive stages, then check overlap.

Hint 2

Terms with r≥2 contain 100.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$11^{8}\equiv1+10(8)\pmod{100}$
Apply the stated relation and retain its conditions.
$R=81$
Reduce the surviving expression modulo 100; it may still exceed 99.

The requested value is 81.

Checks and common pitfalls: Reduce the surviving expression modulo 100; it may still exceed 99.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Defining relation: Use the general term to locate coefficients, constants and coefficient sums.
$(a+b)^n=\sum_{r=0}^nC_n^ra^{n-r}b^r$
Conditions: The exponent n is a nonnegative integer; term number is r+1, not r.