02 September, 2017
Mr Abe said the launch of the missile which flew over Japan was "outrageous", adding: "This is an unprecedented serious and grave threat".
When asked to elaborate, May said there were "a number of issues that China can look at", such as further worldwide sanctions against Kim Jong Un's regime.
Talks over the course of the day focused on North Korea, which on Tuesday fired a missile over Japan, but Brexit was also touched upon.
Britain stands "shoulder to shoulder" with Japan in the face of the increased threat from North Korea, Theresa May has said. These are illegal tests that they have carried out, and they should stop these.
"In response to this illegal action, Japan and the United Kingdom agree to work together to counter this threat, and to work with others in the worldwide community to strengthen pressure against North Korea, including by increasing the pace of sanctions implementation and working towards the adoption of a new and effective resolution at the UNSC", it added.
"I want to express the strong sense of solidarity that the United Kingdom shares with the people of Japan", she added.
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She now enjoys a double-digit lead in the polls against her Social Democratic rival Martin Schulz in her bid for a fourth term. Angela Merkel says she wants to work more closely with Libya's government to curtail illegal migration from Africa to Europe.
May started her visit in Kyoto, the ancient Japanese capital, where she was to attend a traditional tea ceremony and have dinner with Japanese Prime Minister Shinzo Abe.
It is the British leader's first visit to Japan since becoming prime minister in July 2016.
Both Russia and China reiterated their position that engaging in dialogue with the North is the only way to resolve the issue of its continuing nuclear and ballistic missile activities.
Last week, North Korea also threatened Britain with "a miserable end" for taking part in military exercises in South Korea.
But Lord Hague urged Conservative MPs to "get behind" the Prime Minister as she tackles Brexit. But Japanese officials on Tuesday shot down the UK's hopes for a quick trade agreement, saying said they were in no rush to enter negotiations with Britain, especially since it is now negotiating a free trade agreement with the EU.