Can You Find The Unit Digit of 1! +2! +3! +4! +5! +6! +…100! ?

#MathPuzzle #Aptitude #MathTricks

Details : Can You Find The Unit Digit of 1! +2! +3! +4! +5! +6! +…100! ?

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[SimplyLogical]

Can You Find The Unit Digit of "1! +2! +3! +4! +5! +6! +…100!" ?
share your answer.

[rbslife]

Thank you so much sir you help me to clear all my doubts 🙏

[xavierstanton]

Before watching the video:

Let us solve the puzzle with modulo 10 arithmetic. Since 5!=120, 5! is a multiple of 10. But by the definition of the factorial function, all factorial values greater than 5! are a multiple of 120 and thus, a multiple of 10. This means we can factor out a 10 from the sum 5!+6!+...+100!. Hence, 1!+2!+3!+...+100! is congruent to 1!+2!+3!+4! modulo 10. Doing the math reveals that 1!+2!+3!+4! equals 33, which is 3 more than a multiple of 10, meaning 1!+2!+3!+4! is congruent to 3 modulo 10. Congruence in this sense is transitive, which means the original sum is congruent to 3 modulo 10, which makes 3 the last digit.

[erinsgeography]

just a correction: "units" refer to the block of three digits with the lowest value.

For example: 123, 456, 789
there are three blocks (left to right): millions, thousands, units

the furthest to the right are called "units" since the place values on the other blocks are just based of them: hundred thousands, ten thousands, thousands (for example)

Maybe you were thinking about "ones", the place value?
For example: in 123, 456, 789
the "ones" digit is 9

it's called "ones" because it's value is 1 in the base-10 numeral system. The next digit next to it is called the "tens" and the next to the left is called "hundreds"

In 543 for example, 5 is the "hundreds" digit, 4 is the "tens" digit, 3 is the "ones". if you expand 543, you'll have 5(100)+4(10)+3(1), hence the name of the place values

TL;DR: "units" is the block of three digits with the lowest value. "ones" is the place value. saying "units digit" is a paradox, while saying "ones digit" is a tautology. the question could have been better worded as "what digit is in the ones place in the sum of..."

[nageshreddy]

Very good content. keep doing the good work.

[tabindazaheen]

Sir why do we add zero factorial? That's not asked in the question right?

[nurila]

please mr tell me that why you add 0!?

[learnmore]

That mean every situation of factorial case the unit digit is 3