Unit 1 · Topic 09 · Number System & Notation

Forming Numbers When a Digit Repeats

Hook

Ms. Rao's practice sheet said: use four digits, repeat one of them once, build a 5-digit number.

Digits 7, 5, 4, 1 — repeat one digit exactly one time.

Greatest: repeat the biggest digit at the next higher place; smallest: repeat the smallest at the highest place you can.

Watch the Lesson

The Story

Problem one: form the greatest and smallest 5-digit numbers using 7, 5, 4, 1 with one digit repeated once. Anaya sorted descending: 7, 5, 4, 1. For the greatest 5-digit number she placed 7 in the highest place, repeated 7 once (allowed), then 5, 4, 1 — 77,541.

For the smallest she placed the smallest digit 1 first, repeated 1 once, then 4, 5, 7 — 11,457.

Problem two: 6-digit numbers from 8, 7, 5, 2 repeating one digit twice. Greatest: repeat 8 twice at the high places → 88,752. Smallest: start with 2, repeat 2 twice, then 5, 7, 8.

Problem three added 0: digits 9, 6, 1, 0 with one repeat. Greatest: 99,610. Smallest: cannot lead with 0 — place 1 first, repeat 0 once, then 0, 6, 9 → 10,069 (smallest non-zero leads; repeat the smallest usable digit).

Problem four: 8, 5, 2, 0 for a 6-digit number with one digit repeated twice. Greatest: 88,520. Smallest: after 0, the next smallest is 2 — place 2, repeat 0 twice, then 5, 8 → 200,558.

Kabir noticed the pattern: to maximize, repeat the greatest digit at higher places as allowed; to minimize, repeat the smallest digit at the highest place you can — and zero never leads.

6, 6, 2, 9 sorted descending and ascending, repeats included.

So What Just Happened?

When repetition is allowed, sort digits first (descending for greatest, ascending for smallest).

Greatest: place the highest digit at the highest place value; repeat it as many times as permitted before filling lower digits.

Smallest: place the smallest non-zero digit first if zero is present; repeat the smallest allowed digit at higher places when repetition helps minimize.

Zero cannot lead — swap the leading zero with the next smallest non-zero digit.

Examples from practice: 7,5,4,1 once → greatest 77,541, smallest 11,457; with 0 → greatest 99,610, smallest 10,069.

0, 0, 5, 3 sorted for smallest: the non-zero digit still leads, both zeros follow.

Remember This

Sort digits first; repetition limits how many times one digit may repeat.
Greatest: repeat the largest digit at higher places (7,5,4,1 once → 77,541).
Smallest: repeat the smallest digit at the highest place allowed (→ 11,457).
With zero: greatest unchanged; smallest — non-zero leads, repeat smallest/0 as allowed (9,6,1,0 → 10,069).
6-digit with 8,5,2,0 twice: greatest 88,520; smallest 200,558.

Sort every copy of a digit exactly as if it were unique.

Try It Yourself

From 7, 5, 4, 1 with one repeat, write greatest and smallest 5-digit numbers.

From 8, 7, 5, 2 with one digit repeated twice, write greatest and smallest 6-digit numbers.

From 9, 6, 1, 0 with one repeat, write greatest 99,610 and smallest 10,069 and explain the zero rule.

Word Bank

Repeated digit: A digit that appears more than once in the given set.
Descending order: Arranged from largest to smallest.
Ascending order: Arranged from smallest to largest.
Non-zero digit: Any digit from 1 to 9 — anything except 0.

Questions

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