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Divisibility · Topic 3 of 4

📈 Greatest / Least Number

1 exam question type, fully solved

Here you find the largest or smallest number with some divisibility property — the biggest 5-digit multiple of 88, or the least number to add to 888 so it becomes divisible by 35. Every one of these comes from a single division fact.

The one fact behind all of these

Every division can be written as , where is the divisor, the quotient and the remainder (). Read off whatever the question wants:

• Largest N-digit multiple of d: take the largest N-digit number, divide by d, and subtract the remainder. (Smallest multiple: take the smallest N-digit number and add .)

• Least number to SUBTRACT to make N divisible: that is just the remainder .

• Least number to ADD to reach the next multiple: that is .

• Several divisors at once: replace them with their LCM, then it is a normal greatest/least-multiple question.

• If a remainder is required (e.g. “leaves remainder 3”): make it exactly divisible first, then add that remainder back at the end.

The classic trap is to give the remainder when the question asked you to ADD — remember, you add , not .

🔁Remainder ExplorerSee how a remainder is built
Dividend
Divisor
100=7 × 14+2
7
7
7
7
7
7
7
7
7
7
7
7
7
7
2
Remainder = 2
Least number to subtract for divisibility = 2 ·  least to add = 5

Since , the remainder is 13, so the least number to add to reach the next multiple of 35 is .

📝Practice Questions

Q1The largest 3-digit number divisible by 7 is:

Q2The least number to add to 123 to make it divisible by 8 is:

Q3The smallest number divisible by both 12 and 18 is:

📚

Real exam questions — Greatest / Least Number

1 question types · 5 solved examples from real SSC papers

It all comes from N = divisor × quotient + remainder. Add (divisor − remainder), subtract the remainder, and use the LCM when several divisors apply.

How to solve this type
Everything rests on the division fact N=(divisor)×q+rN = (\text{divisor})\times q + r, where r is the remainder. To find the largest N-digit multiple, take the largest N-digit number and subtract its remainder. To make a number divisible, the least number to SUBTRACT is the remainder r; the least number to ADD is (divisor − r). When several divisors apply at once, first replace them with their LCM. If a remainder is required, make it exactly divisible first, then add the remainder back at the end.

What is the smallest number that must be added to 888 to make it divisible by 35?

A23B21C20D22

What is the largest 5-digit number that is exactly divisible by 88?

A99984B99986C99968D99992

What is the least number that must be added to 10000 to make it divisible by 327?

A137B190C327D237

What is the greatest number less than 10000 that is divisible by 16, 21, 24 and 28?

A9408B9072C9744D9576

What is the largest 5-digit number that leaves remainder 3 when divided by 7, 9 and 11?

A96720B99795C95840D98685