What is the units digit of the 5857th triangular number.?

Answer 1

The nth triangular number is given by ½ × n × (n+1)

→ the 5857th triangular number is ½ × 5857 × 5858 = 17,155,153, so its units digit is a 3.

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Alternatively,

If you look at the units digits of the first 20 triangular numbers they are {1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0}

At this stage, as we are only concerned with the units digit, as we now have a 0 for the units digit, when 21 is added it is the same as adding 1 to 0 to give a 1, for the 22nd triangular number, we are adding 2 to the 1 to give 3, and so on - the sequence of 20 digits is repeating.

To find the units digit of the nth triangular number, find the remainder of n divided by 20 and its units digit will be that digit in the sequence (if the remainder is 0, use the 20th number). To find the remainder when divided by 20 is very simple by looking at only the tens digit and the units digit:

• If the tens digit is even (ie one of {0, 2, 4, 6, 8}), the remainder is the units digit
• If the tens digit is odd (ie one of {1, 3, 5, 7, 9}), the remainder is the units digit + 10.

5857 ÷ 20 = ... remainder 17; the 17th digit of the above sequence is a 3, so the units digit of the 5857th triangular number is a 3.

This trick can be used for much larger triangular numbers which calculators cannot calculate exactly using the above formula. eg the units digit of the 1234567890123456789th triangular number is... 1234567890123456789 ÷ 20 = .... remainder 9, so this triangular number's units digit is the 9th digit of the above sequence which is a 5.