Consider the multiplication 999 × abc = def132 in decimal notation, where a, b, c, d, e and f are all single digits and abc and def represent three digit numbers. What are the values of a, b, c, d, e and f respectively?

Introduction / Context:
This question involves a neat property of multiplying a three digit number by 999. The product 999 × abc is written as def132, where both abc and def are three digit numbers and all letters represent individual digits. We must deduce the digits a, b, c, d, e and f using an algebraic identity and place value reasoning.

Given Data / Assumptions:
We have 999 × abc = def132.abc and def are three digit numbers made of digits a, b, c and d, e, f respectively.The last three digits of the product are 1, 3 and 2 in that order.All letters represent decimal digits 0 to 9, with a and d non zero.

Concept / Approach:
Using the identity 999 = 1000 − 1, we can write 999 × abc as 1000 × abc − abc. This means that 999 × abc is equal to the number obtained by writing abc000 and then subtracting abc. For any three digit number abc, this subtraction has a special pattern: the product 999 × abc can be written as the concatenation of (abc − 1) followed by (1000 − abc), each part being three digits.

Step-by-Step Solution:
Step 1: Let N be the three digit number abc.Step 2: Compute 999 × N as (1000 − 1) × N = 1000N − N.Step 3: In decimal notation, 1000N is N followed by three zeros, and subtracting N gives a six digit number that can be viewed as (N − 1)(1000 − N), where both parts are written with three digits (including leading zeros if necessary).Step 4: In the given product 999 × abc = def132, the last three digits are 132, so 1000 − N must be 132.Step 5: Solve for N: 1000 − N = 132 implies N = 1000 − 132 = 868.Step 6: So abc = 868, meaning a = 8, b = 6 and c = 8.Step 7: The first three digits def must equal N − 1, so def = 868 − 1 = 867.Step 8: Therefore d = 8, e = 6 and f = 7.

Verification / Alternative check:
Verify directly by multiplication: 999 × 868 = (1000 − 1) × 868 = 868000 − 868. Subtracting, we get 867132. This matches the pattern def132 with def = 867 exactly. The digits are therefore (a, b, c, d, e, f) = (8, 6, 8, 8, 6, 7).

Why Other Options Are Wrong:
Each wrong option assigns a different order to the digits 6, 7 and 8. Substituting these into the pattern would not satisfy the relation 1000 − abc = 132 or would give a product different from def132. Only the combination 8, 6, 8, 8, 6, 7 simultaneously satisfies the identity and the final three digits requirement.

Common Pitfalls:
Some learners try to brute force the multiplication with each option instead of using the 1000 − 1 identity, which is more time consuming. Others may misapply the concatenation idea or miscalculate 1000 − N. Carefully applying the identity 999N = (N − 1)(1000 − N) provides a fast and reliable path to the answer.

Final Answer:
The correct digits are 8, 6, 8, 8, 6, 7 for (a, b, c, d, e, f) respectively.