Digit sums of successive integers
For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
Keep in mind that $0 notin mathbb{N}$.
mathematics no-computers number-theory
asked 3 hours ago
A. P.A. P.
3,47411144
add a comment |
For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
Keep in mind that $0 notin mathbb{N}$.
mathematics no-computers number-theory
asked 3 hours ago
A. P.A. P.
3,47411144
add a comment |
For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
Keep in mind that $0 notin mathbb{N}$.
mathematics no-computers number-theory
asked 3 hours ago
A. P.A. P.
3,47411144
For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
Keep in mind that $0 notin mathbb{N}$.
mathematics no-computers number-theory
mathematics no-computers number-theory
asked 3 hours ago
A. P.A. P.
3,47411144
asked 3 hours ago
A. P.A. P.
3,47411144
asked 3 hours ago
A. P.A. P.
3,47411144
asked 3 hours ago
A. P.A. P.
3,47411144
asked 3 hours ago
A. P.A. P.
3,47411144
3,47411144
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
69999 (42) and 70000 (7)
...
No two consecutive integers are both multiples of 7, so this needs to take place at a rollover.
...
Rolling over a single 9 drops the digit sum by 8, which isn't a multiple of 7 either.
...
Similarly, if Y=X+1, X99->Y00 drops by 17 and X999->Y000 drops 26.
...
X9999 to Y0000 is the first drop (35) which is itself a multiple of 7...
...
Any number of 9's that's congruent mod 7 to 4 will work, but they'll be much larger, so the first instance must roll over 4 9's.
...
From there, all that remains is to find the first multiple of 10000 with an appropriate digit sum.
...
My initial, less confidence-inspiring method just recognized that 7*10^n was a likely candidate for x+1, and so I started appending 9's to a single 6 until the sum worked out...
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
add a comment |
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
69999 (42) and 70000 (7)
...
No two consecutive integers are both multiples of 7, so this needs to take place at a rollover.
...
Rolling over a single 9 drops the digit sum by 8, which isn't a multiple of 7 either.
...
Similarly, if Y=X+1, X99->Y00 drops by 17 and X999->Y000 drops 26.
...
X9999 to Y0000 is the first drop (35) which is itself a multiple of 7...
...
Any number of 9's that's congruent mod 7 to 4 will work, but they'll be much larger, so the first instance must roll over 4 9's.
...
From there, all that remains is to find the first multiple of 10000 with an appropriate digit sum.
...
My initial, less confidence-inspiring method just recognized that 7*10^n was a likely candidate for x+1, and so I started appending 9's to a single 6 until the sum worked out...
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
add a comment |
69999 (42) and 70000 (7)
...
No two consecutive integers are both multiples of 7, so this needs to take place at a rollover.
...
Rolling over a single 9 drops the digit sum by 8, which isn't a multiple of 7 either.
...
Similarly, if Y=X+1, X99->Y00 drops by 17 and X999->Y000 drops 26.
...
X9999 to Y0000 is the first drop (35) which is itself a multiple of 7...
...
Any number of 9's that's congruent mod 7 to 4 will work, but they'll be much larger, so the first instance must roll over 4 9's.
...
From there, all that remains is to find the first multiple of 10000 with an appropriate digit sum.
...
My initial, less confidence-inspiring method just recognized that 7*10^n was a likely candidate for x+1, and so I started appending 9's to a single 6 until the sum worked out...
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
add a comment |
69999 (42) and 70000 (7)
...
No two consecutive integers are both multiples of 7, so this needs to take place at a rollover.
...
Rolling over a single 9 drops the digit sum by 8, which isn't a multiple of 7 either.
...
Similarly, if Y=X+1, X99->Y00 drops by 17 and X999->Y000 drops 26.
...
X9999 to Y0000 is the first drop (35) which is itself a multiple of 7...
...
Any number of 9's that's congruent mod 7 to 4 will work, but they'll be much larger, so the first instance must roll over 4 9's.
...
From there, all that remains is to find the first multiple of 10000 with an appropriate digit sum.
...
My initial, less confidence-inspiring method just recognized that 7*10^n was a likely candidate for x+1, and so I started appending 9's to a single 6 until the sum worked out...
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
69999 (42) and 70000 (7)
...
No two consecutive integers are both multiples of 7, so this needs to take place at a rollover.
...
Rolling over a single 9 drops the digit sum by 8, which isn't a multiple of 7 either.
...
Similarly, if Y=X+1, X99->Y00 drops by 17 and X999->Y000 drops 26.
...
X9999 to Y0000 is the first drop (35) which is itself a multiple of 7...
...
Any number of 9's that's congruent mod 7 to 4 will work, but they'll be much larger, so the first instance must roll over 4 9's.
...
From there, all that remains is to find the first multiple of 10000 with an appropriate digit sum.
...
My initial, less confidence-inspiring method just recognized that 7*10^n was a likely candidate for x+1, and so I started appending 9's to a single 6 until the sum worked out...
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
edited 2 hours ago
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
answered 2 hours ago
ZomulgustarZomulgustar
1,728622
1,728622
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
add a comment |
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
As this is a 'no-computers' puzzle, could you elaborate on how you find this number? Most likely you will also see whether it's minimal if you go through these steps.
– A. P.
2 hours ago
add a comment |
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