Using all digits from to exactly once (the first digit cannot be ) to create a -digit number, write the — (a) Largest multiple of (b) Smallest even number

Question

Homi · Accepted Answer

Understand the Question

We need to form a $10$ -digit number using each digit from $0$ to $9$ exactly once, where the first digit cannot be $0$ .
To form the largest number, place the largest available digits in the highest place values (from left to right).
To form the smallest number, place the smallest non-zero digit ( $1$ ) first, followed by the smallest available digits (including $0$ ) in ascending order.
Divisibility Conditions:
- A multiple of $5$ must end with $0$ or $5$ .
- An even number must end with an even digit: $0, 2, 4, 6,$ or $8$ .

(a) Largest multiple of $5$

Step 1 · Determine the Largest Multiple of 5

To maximize the number, arrange the digits in descending order: $9, 8, 7, 6, 5, 4, 3, 2, 1, 0$ .

A multiple of $5$ must end in $0$ or $5$ :

If the units digit is $0$ : $9876543210$
If the units digit is $5$ : $9876432105$

Comparing the two numbers: $9876543210 > 9876432105$

Therefore, the largest multiple of $5$ is $\text{9,876,543,210}$ .

Answer

(a) $\text{9,876,543,210}$

(b) Smallest even number

Step 1 · Determine the Smallest Even Number

To minimize the $10$ -digit number, the first digit must be $1$ , and the subsequent place values should be as small as possible in ascending order. The units digit must be an even number ( $0, 2, 4, 6, 8$ ).

Case 1: Last digit is $0$ Remaining digits arranged in ascending order: $2, 3, 4, 5, 6, 7, 8, 9$ $1234567890$

Case 2: Last digit is $2$ Remaining digits arranged in ascending order: $0, 3, 4, 5, 6, 7, 8, 9$ $1034567892$

Case 3: Last digit is $4$ Remaining digits arranged in ascending order: $0, 2, 3, 5, 6, 7, 8, 9$ $1023567894$

Case 4: Last digit is $6$ Remaining digits arranged in ascending order: $0, 2, 3, 4, 5, 7, 8, 9$ $1023457896$

Case 5: Last digit is $8$ Remaining digits arranged in ascending order: $0, 2, 3, 4, 5, 6, 7, 9$ $1023456798$

Comparing all cases: $1023456798 < 1023457896 < 1023567894 < 1034567892 < 1234567890$

Therefore, the smallest even number is $\text{1,023,456,798}$ .

Answer

(b) $\text{1,023,456,798}$

Common Mistakes

Leading Zero Error: Placing $0$ in the first position makes it a $9$ -digit number instead of a $10$ -digit number.
Suboptimal Last Digit Selection: Picking $0$ as the units place in part (b) prevents $0$ from being in the second position (hundred-millions place), resulting in $1234567890$ , which is much larger than reserving $8$ for the units place to get $1023456798$ .

Question 13

More questions in FIO