Let x = 62357 - 2730, then what is the unit’s digit of x?

Concept

The unit (last) digit of a power depends only on the unit digit of the base, and that unit digit repeats in a fixed cycle as the exponent grows. For a base ending in 3 the cycle is 3, 9, 7, 1 (length 4); for a base ending in 7 the cycle is 7, 9, 3, 1 (length 4). To get the unit digit of a difference, find the unit digit of each term, then subtract — borrowing when the first term's unit digit is smaller.

Application

1. 62357: the base ends in 3, so use the cycle 3, 9, 7, 1 of length 4. Divide the exponent: 57 ÷ 4 leaves remainder 1, so the unit digit is the 1st entry of the cycle = 3.

2. 2730: the base ends in 7, so use the cycle 7, 9, 3, 1 of length 4. Divide the exponent: 30 ÷ 4 leaves remainder 2, so the unit digit is the 2nd entry of the cycle = 72 = 49, i.e. 9.

3. Subtract the unit digits: a number ending in 3 minus a number ending in 9. Since 3 is smaller than 9, borrow 1 from the tens place: 13 − 9 = 4.

Cross-check

Computing the last digits directly: 62357 ends in 3 and 2730 ends in 9; their difference ends in …54. So the unit digit of x is 4.